title: Electrostatic energy integral for point charges

text: The electric energy stored in a system of two point charges $Q_1$ and $Q_2$ is simply $$W = \frac{1}{4\pi\epsilon_0}\frac{Q_1Q_2}{a}$$ where $a$ is the distance between them. However, the total energy can also be calculated through the volume integral of magnitude squared of the electric over all space: $$W = \frac{\epsilon_0}{2}\int_{\mathbb{R}^3} E^2\,dV$$ Suppose that $Q_1$ sits on the origin and $Q_2$ a distance $a$ away on the $z$-axis. Then the electric field is $$\vec{E}(x,y,z) = \frac{1}{4\pi\epsilon_0}\left[\left(\frac{Q_1}{r^3} + \frac{Q_2}{d^3}\right)x\hat{x} +\left(\frac{Q_1}{r^3} + \frac{Q_2}{d^3}\right)y\hat{y} + \left(\frac{Q_1z}{r^3} + \frac{Q_2\left(z-a\right)}{d^3}\right)\hat{z}\right]$$ where \begin{align} r &= \sqrt{x^2+y^2+z^2} \\\ d&= \sqrt{x^2+y^2+\left(z-a\right)^2} \end{align} Calculating $W$ through $\int_{\mathbb{R}^3} E^2\,dV$ seems extremely difficult; Mathematica, for example, appears stumped. Yet its result should simply be $\frac{1}{4\pi\epsilon_0}\frac{Q_1Q_2}{a}$, correct? The integral formula still applies to point charges, correct?


title: Why doesn't Bernoulli's Principle apply to Current and Resistors in a circuit?

text: Bernoulli's principle makes sense when you apply it to fluids. If you decrease the diameter of a pipe then the velocity of the fluid increases because it needs to keep the same rate of fluid moving through the pipe. So my question is: If Voltage == Diameter of the pipe and Current == Rate of which the fluid is moving Why do resistors work? Shouldn't the resistor only actually work within itself but then return the current to it's actual rate once you have passed it? Or have I taken the analogy of wires being like pipes of water to far?


title: design of the heat exchanger...in chimney

text: ![heat exchanger diagram](http://i.stack.imgur.com/w04j8.jpg) I want to design a heat exchanger in a chimney in order to utilize heat from chimney. I have done several experiments, but I could not determine the exact length of tube (carrying water), such that its inlet temp is ambient temperature and outlet temperature is expected to be 100 degree Celsius. I would be very thankful if some one would help me determine the length of the tube. tube diameter is 10mm (almunium), chimney diameter is(120 mm). Can anyone help me with formulas involved in calculating the heat transfer and length of the tube... I already calculated the lenght experimentally but I could not do it mathematically.


title: Why does a mirror reverse polarization of circularly polarised light?

text: A glass mirror (with metal backing layer) will reverse the polarisation of circularly polarised light upon reflection. A polished piece of metal will also reverse the polarisation of circularly polarised light upon reflection. (I have tested and confirmed this for myself). wikipedia states the reason a mirror will reverse the polarisation of circularly polarised light is: > ...[A]s a result of the interaction of the electromagnetic field with the > conducting surface of the mirror, both orthogonal components are effectively > shifted by one half of a wavelength. However, my understanding of mirrors is that only a polished piece of metal will phase shift a wavelength by half a wavelength, whereas a glass mirror (with metal backing layer) will not produce a phase shift. For example wikipedia which states: > According to Fresnel equations there is only a phase shift if n2 > n1 (n = > refractive index). This is the case in the transition of air to reflector, > but not from glass to reflector


title: Why do they call it quantum teleportation?

text: So I have been trying to learn about entanglement and quantum teleportation and from what I've been able to gather so far, the **teleportation** part seems to be misleading. At first I thought that the two particles were of a uniform undetermined state which would collapse probabilistically upon observation. But the more I read into it, it began to seem like the following unimpressive scenario as it relates to the Schrödinger's cat analogy: It seems as though a dead cat and a live one are put into two boxes that can only be observed once. When one person gets the live cat, they know the other got the dead cat. They can infer this knowledge instantly, but who cares because it took UPS two days to ship the thing. If this is the case, why do they call it _quantum teleportation_ and if it isn't the case could someone explain it to me better?


title: What is Dalitz decay?

text: What is Dalitz decay? I know there are Dalitz $\pi^0 \to e^+ + e^- + \gamma$ decay, $w \to \pi^0 + e^+ + e^-$ decay, may be more. But is there a rule to say which decay is Dalitz and which is not? Is there a rule to say which particle can decay by Dalitz decay and which does not?


title: Pockels effect/Kerr effect

text: In Pockels electro-optic effect, change in Electric field produces change in refractive index/ birefringence. Moreover this effect becomes electric field squared in the case of Kerr effect. If we increase the electric field more for example 3 ,4 , 5 times . we will observe the same phenomenon of Kerr effect or is it something else ?


title: Potential energy: Electric field two spherical charges

text: I want to determine the potential energy of two equally charged spherical charges by using the equation: $V_{pot}= \int_V \frac{1}{2} \epsilon_0 E^2 dV$ and therefore I was wondering what I has to take as $E$? The sum of the electric fields of both charges? Notice, that I want to use this equation on purpose since it enables me to determine the potential energy in a particular volume.


title: Tighten rope around cylindricaly shaped space

text: Imagine we live on cylinder(we are 2d creature), put a rope around that cylinder and start pulling both ends of the rope against each another. Will the space get deformed? I guess it will, I have to put some energy to the rope and energy deforms the space. Is it possible to calculate shape of the space based on the force I put in the rope? In order to calculate this I guess you have to make 3d analogy and use laws of general relativity.


title: Full calculation of B meson mixing amplitude

text: I am trying to calculate B mixing in the Standard Model (in preparation to go beyond the SM). I have no trouble doing the gamma matrix algebra etc. but the loop integral keeps tripping me up. In my calculation I have $$ \int \frac{d^4 k}{(2\pi)^4} \frac{k^2}{(k^2-m_1^2) (k^2 - m_2^2) (k^2- M_W^2)^2} $$ I know about Feynman parametrization etc. but the result I get does not comply with what I find in the Literature. Unfortunately basically all calculations simply say "we calculate by standard methods" and have a function $$S(x_t) = \frac{4x_t - 11 x_t^2 + x_t^3}{4(1-x_t)^2} - \frac{3x_t^3 \ln x_t}{2(1-x_t)^3} $$ with $x_t = m_t^2 / M_W^2$ if $m_1 = m_2 = m_t$. This is not directly the result of evaluating the above integral though, since I have at least a factor of $1/M_W^2$ that is in the integral, but not included in S. Where can I find a full calculation of the box diagram and what is the exact relation of the loop integrals to the functions $S$? I do know about the general 1, 2, 3 and 4 point functions, generally called $A_0, B_0, B_1, \dots$. The $S$ is different from $D(0, 0, 0, m_1, m_2, M_W, M_W)$ though!


title: What really goes on in a vacuum?

text: I've been told that a vacuum isn't actually empty space, rather that it consists of antiparticle pairs spontaneously materialising then quickly annihilating, which leads me to a few questions. Firstly, is this true? And secondly, if so, where do these particles come from?... (do the particles even have to come from anywhere?)


title: How do I calculate motor efficiency from voltage, current and RPM?

text: I have a setup where a motor is spinning at a constant (known) RPM, under no load. I know the power going into the motor (voltage * current), and I can find out the rotational kinetic energy of the rotor. My question is, how do I calculate the efficiency of the motor from this? Is it even possible? I understand that the motor is not doing any "useful work", but it is still overcoming frictional forces to keep spinning, and there are definitely energy losses there.


title: Shooting a single photon through a double slit

text: Consider the image below. It shows a double slit experiment but with a single photon at a time. **My question is as follows** : Why is it that the photons always take a different path when shot at the same target? Where does the uncertainty lie? If we shoot it in exactly the middle of the two slits, why does it have a 50-50 chance of going into either slit? And why is the amount of diffraction for a single photon always different? I know that people will say that the photon enters both the slits at the same time and things like that. But does anyone have an intuitive explanation as to why this happens? Why does a photon, shot with the same frequency and exactly in the same direction, still have the probability of entering either slit. Why is it that the diffraction of a single photon is different for the same wavelength? So in brief, why does a single photon which is shot in an exactly the same why as the previous one, end up being in a different place. I am looking forward to an answer with the least possible math (if any). Or is it that a photon cannot be shot in the exact same way as the previous one? ![enter image description here](http://i.stack.imgur.com/pshsu.gif) Image source: http://abyss.uoregon.edu/~js/images/photon_double_slit2.gif


title: Maxwell's equations in microscopic and macroscopic forms, and quantization

text: The macroscopic Maxwell's equations can be put in terms of differential forms as $$\mathrm{d}\mathrm{F}=0,\quad\delta \mathrm{D}=j\implies \delta j=0,\quad \mathrm{D}=\mathrm{F}+\mathrm{P}.$$ $\mathrm{F}$, $\mathrm{D}$, and $\mathrm{P}$ are 2-forms; $j$ is a 1-form. $\mathrm{F}$ is the $electromagnetic\ field$, which has 3-vector components $E_i,B_i$, usually called the $electric\ field$ and the $magnetic\ field$; $\mathrm{D}$ is a generalized electromagnetic field, generalized in the sense that $\mathrm{dD}\not=0$, which has 3-vector components $D_i,H_i$, called, amongst other names, the $electric\ displacement\ field$ and the $magnetizing\ field$; $j$ is a conserved 4-current; and $\mathrm{P}$ is the electromagnetic polarization/magnetization, which has 3-vector components $P_i,M_i$, usually called, up to a factor $\pm 1$, the $polarization$ and the $magnetization$, which in the macroscopic form of Maxwell's equations model the internal degrees of freedom, whereas in the microscopic form of Maxwell's equations we have $\mathrm{P}=0$. The Maxwell's equation Wiki page, together with its many links, does a reasonable job of summarizing all this. In QED, there are internal degrees of freedom associated with the Dirac field, so should we be working with $\mathrm{D}$ when we quantize? Classically, given a differentiable bivector field such as $\mathrm{D}$, there is inevitably a conserved 4-current $\delta \mathrm{D}$, unless the equations of motion are such that this quantity is identically zero. [Edit, to reflect Cristi Stoica's comment] Given an arbitrary 2-form such as $\mathrm{D}$, _if_ Hodge decomposition could be applied, we could decompose $\mathrm{D}$ uniquely as $$\mathrm{D}=\mathrm{dA}+(\mathrm{D}-\mathrm{dA}).$$ For Minkowski space, such decompositions are possible but are _not_ unique because there is no positive- definite bilinear form on the function space; for the forward light-cone component $\mathrm{D}^+$ the decomposition can be restricted by minimizing the positive semi-definite integral $$\int [\widetilde{\delta(\mathrm{D}-dA)}]_\mu(k)[\widetilde{\delta(\mathrm{D}-dA)}]^\mu(k)\theta(k^2)\theta(k_0)\frac{\mathrm{d}^4k}{(2\pi)^4},$$ and similarly for the backward light-cone, but no such restriction is available for space-like components. Looked at this way, the electromagnetic field $\mathrm{F}$ is just a component of the real object of interest, $\mathrm{D}$, that can be expressed as $\mathrm{F}=\mathrm{dA}$, for which, inevitably, $\mathrm{dF}=0$. The nonunique component $\mathrm{D}-\mathrm{dA}=\mathrm{P}$ of $\mathrm{D}$ cannot be expressed in terms of a single potential. [End of edit -- constructed somewhat in the moment so it may have to be modified.] If $\mathrm{D}$ is taken to be the real object of interest, with $\delta\mathrm{D}=j$ an independent degree of freedom, then does that mean that we should look for second-order differential equations for $\mathrm{D}$ in the first instance, probably in association with additional degrees of freedom, in view of the additional degrees of freedom that are present in the Standard Model? I should point out that, for the purposes of this question, I take the Dirac wave function not to be observable in the classical Maxwell-Dirac theory, with $\overline{\psi(x)}\psi(x)$, $(\delta\mathrm{D})^\mu=j^\mu=\overline{\psi(x)}\gamma^\mu\psi(x)$, $\overline{\psi(x)}\gamma^{[\mu}\gamma^{\nu]}\psi(x)$, $\overline{\psi(x)}\gamma^{[\mu}\gamma^\nu\gamma^{\rho]}\psi(x)$, and $\overline{\psi(x)}\gamma^5\psi(x)$ being (non-independent) 0-, 1-, 2-, 3-, and 4-form observables, respectively. This is almost not a question, for which apologies, however I am interested in any critique of details or of whatever minimally elaborated plan there might seem to be to be here, or in any reference that seems pertinent. The ideas lie somewhat in the same territory as an earlier question of mine, Gauge invariance for electromagnetic potential observables in test function form, in that I prefer to disavow unobservable fields unless there are absolutely clear reasons why they are unavoidable.


title: What has been measured at the Higgs experiment and what do we know now?

text: Explained at the level of a 5$^{\text {th}}$ semester physics student (i.e. pre QFT, but far beyond the level of a news article for non-physicists, which avoids all details and only deals in analogies) ...   * What has been measured at CERN some days ago?   * What are the essential ingredients of the theory necessary to interpret said measurement? And so how do we deduce from the results that there is a new field/particle observed?   * How to read the most relevant graphs in the presentation of the results?


title: Neutron decay and electron anti neutrino $n\to p + e + \bar{\nu}_e$

text: Why do we need neutrino to explain neutron decay? Is there any evidence regarding existence neutrinos in the context of $n\to p + e + \bar{\nu}_e$?


title: What limits the maximum sustainable surface charge density of a sphere in space?

text: Suppose I charge a sphere and leave it in vacuum for 10 years. After that time, I want its surface charge density to be in the order of 10^5C/m^2. Would that be possible? Would it depend on the material used and how? Would adding or removing electrons make a difference (positively vs negatively charged)?


title: Definitions of Lagrange points: $L_4$ and $L_5$

text: We have the the five Lagrange points (let consider Earth and Sun):   * $L_1$ - lie between Sun and Earth;   * $L_2$ - beyond the Earth;   * $L_3$ - beyond the Sun; And what's the difference between $L_4$ and $L_5$? Does they defined with the respect to rotation of Earth around the Sun?


Why does the nature always prefer low energy and maximum entropy? I've just learned electrostatics and I still have no idea why like charges repel each other. http://in.answers.yahoo.com/question/index?qid=20061106060503AAkbIfa I don't quite understand why U has to be minimum. Can someone explain?


title: Simple explanation of Quantum Zeno Effect

text: I'm a student and I had to give a talk on seminar about Quantum Zeno effect and Anti-Zeno effect to my colleagues (all listeners have had a course in quantum physics, but not a heavy one with all the bra and ket stuff). My first idea to give a simple explanation of Zeno's effect was this: Let's take a look at exponential decay where the chance for particle or state to survive some time $t$ is $P_S=e^{-t/\tau}$. If I measure it after time $\tau$ I have chance $P_S=1/e$ that it will still be intact. If I instead allow it to do it's things only for time $\tau/N$ and then measure it, the survival chance will be $P_S=e^{-1/N}$ which approaches $1$ as $N$ increases. To achieve the same total time, I have to repeat this procedure $N$ times and the total survival probability is... $P_S=(e^{-1/N})^N=1/e$. So it obviously doesn't work, I get no Zeno's effect in this way. It's interesting that after I gave the talk professor rose and said "Well, this can be easily understood if we look at the exponential decay". Then he started drawing exponent and another exponent that's repeatedly interrupted and reset to initial state after small intervals. Later we agreed that this won't actually work, but the question is - why? Why doesn't this intuitively obvious way doesn't work and what would be the correct law out of which one could see the Zeno's effect? Is there any elegant way to explain this effect without heavy math and angles of state vectors? ADDITIONAL QUESTION (related): Is it correct to use name "Quantum Zeno Effect" for turning of polarization by series of inclined polarizers or the thing that is done in this article?


title: Gaussian type integral with negative power of variable in integrand

text: How can we compute the integral $\int_{-\infty}^\infty t^n e^{-t^2/2} dt$ when $n=-1$ or $-2$? It is a problem (1.11) in Prof James Nearing's course _Mathematical Tools for Physics._ Can a situation arise in physics where this type of integral with negative power can be used?


title: Linear size of everything in the Universe is doubled overnight

text: Suppose the linear size of everything is doubled overnight. Can you test the statement by measuring sizes by a meter stick? Can you test it by using the fact that the speed of light is a universal constant and has not changed? What will happen if all the clocks in the universe start running at half speed?


title: An alternative definition of the creation and annihilation operators?

text: Suppose we have a system of bosons represented by their occupation numbers $$\tag{1} | n_1, n_2, ..., n_\alpha, ... \rangle$$ Then we can define creation and annihilation operators $$\tag{2} a_\alpha^\dagger| n_1, n_2, ..., n_\alpha, ... \rangle = \sqrt{n_\alpha+1} | n_1, n_2, ..., n_\alpha+1, ... \rangle$$ $$\tag{3} a_\alpha| n_1, n_2, ..., n_\alpha, ... \rangle = \sqrt{n_\alpha} | n_1, n_2, ..., n_\alpha-1, ... \rangle$$ This is nice because the number operator is just $a_\alpha^\dagger a_\alpha$. However, would it be sensible to define an alternate set of operators to work with? $$\tag{4} b_\alpha| n_1, n_2, ..., n_\alpha, ... \rangle = | n_1, n_2, ..., n_\alpha+1, ... \rangle$$ $$\tag{5} c_\alpha| n_1, n_2, ..., n_\alpha, ... \rangle = \begin{cases} | n_1, n_2, ..., n_\alpha-1, ... \rangle & n_\alpha>0 \\\ 0 & n_\alpha=0 \end{cases}$$ $$\tag{6} N_\alpha| n_1, n_2, ..., n_\alpha, ... \rangle = n_\alpha| n_1, n_2, ..., n_\alpha, ... \rangle $$ Why don't we work with these operators? The bosonic creation and annihilation operators $a_\alpha^\dagger$ and $a_\alpha$ were defined to mimic the harmonic oscillator's raising and lowering operators ($x \pm i p$), but is there any compelling reason to keep the $\sqrt{n_\alpha+1}$ and $\sqrt{n_\alpha}$ factors? I suppose $a_\alpha^\dagger$ and $a_\alpha$ obey nice properties such as $[a_\alpha,a_\alpha^\dagger]=1$ and the fact that they are Hermitian adjoints of each other. What are the analogous relationships that $b_\alpha$ and $c_\alpha$ would obey?


title: Construction of a spin chain Hamiltonian invariant under a finite subgroup of SO(3)

text: I would like to construct a 2-local Hamiltonian that acts on a 1D spin chain where each spin transforms as the 3D irrep of $A_4$ which is a subgroup of $SO(3)$. I know that an $SO(3)$ invariant Hamiltonian can be constructed using the Casimir operator like $H = \sum_i \vec{S_i} . \vec{S_{i+1}}$. Is there a similar thing for finite groups (i.e equivalent of a Casimir)? I would explicitly like the on site symmetry to be only $A_4$ and nothing bigger. (eg: I cannot just use $H = \sum_i \vec{S_i} . \vec{S_{i+1}}$). I also need parity and translation invariance but I know how to impose those.


title: Is there an efficiency-optimizing layout of items in a refrigerator?

text: Is there some optimal layout of items, or types of items (characterized by size, shape, density, solid/liquid, etc) that will allow a common household refrigerator to function with optimal energy efficiency? I know this is a very broad and any answer will have to make several assumptions, not the least of which include how often the refrigerator is opened and for how long. I'd be perfectly happy with a "steady-state" answer that assumes, e.g. the refrigerator is never opened. But more is always better. If nothing else, I'm curious about what considerations might go into conducting a more thorough analysis.


title: How come these 3d glasses don't show a difference between horizontal and vertical?

text: I thought that modern 3d glasses work by having one lens filter horizontally polarized light, and the other filter vertically polarized light. However, I found this pair of 3d glasses at my parents' house, and looked at the reflection from the floor at different angles: ![](http://i.stack.imgur.com/GqauM.jpg) ![](http://i.stack.imgur.com/hA3oB.jpg) What's confusing me is that turning the glasses 90 degreed changed the color of the light from yellow to blue, but it did that on both lenses simultaneously. I expected one to be yellow while the other is blue, and vice versa, since they should be polarized at a 90 degrees difference. Can someone explain this?


title: Why is it possible to expand the SM Higgs field in its second component only

text: In the lecture, the professor said something along the lines of: "After a suitable gauge transformation, the standard model higgs field can be expanded as $$\phi =\left(\begin{array}{c} 0 \\\ v+H(x) \end{array}\right)$$ ". Now, the argument I have been able to scramble from different sources goes along these lines:   1. We can write small higgs field excitations as $$\tilde\phi =\left(\begin{array}{c} i \theta_1(x) + \theta_2(x) \\\ v+H(x) - i \theta_3(x) \end{array}\right)$$   2. An appropriately chosen local $SU(2)_L$ transformation transforms this into the above form where all $\theta=0$.   3. Thus, by applying an appropriate local $SU(2)_L$ transformation to all elements of the Lagrangian, we can use this form without loss of generality. I have the following two issues with this argumentation: First, how do I know that the first statement is true? Secondly and most importantly, wouldn't any symmetriy transformation that corrects the higgs weak spinor into the above from , given a random but fixed excitation of the higgs field, also mess up the spinors of the left handed leptons? In other words, how is it possible to find a local gauge transformation on the Lagrangian that corrects any $\tilde \theta$ into $\theta$, but does not change the form of $ \bar L =\left(\begin{array}{c} \bar\nu \\\ \bar e \end{array}\right)$ ??


title: Is the flatness of space a measure of entropy?

text: This is a bit quirky: For a very long time I've found Stephen Hawking's evaporating small black holes a lot more reasonable and intuitive than large black holes. The main reason is that gravity is relative only if your gravity vectors are all parallel. When that is true, you can simply accelerate along with the field and have a perfectly relativistic frame going for you. Not so if your gravity vectors angle in towards each other, as is particularly true for very small black holes. In that case the energy inherent in the space around the hole becomes quite real and quite hot, and that's regardless of whether you have matter in the mix or not. (Hopefully that's fairly intuitive to everyone in this group?) So, how can the space immediately surrounding a tiny black hole _not_ be incredibly hot? By its very geometry it must be absolutely bursting with energy due to the non-parallel intersection of extremely intense gravity vectors. So, the idea of that energy evidencing itself in the creation of quite real particles _outside_ of the event horizon seem almost like a necessity, a direct consequence of the energetic structure of space itself. So, that's really the basis for my question: Isn't the _curvature_ of space a better way to understand its entropy adding up the surface area of a black hole? By focusing on curvature, _all_ space has entropy, not just the peculiar variety of space found on event horizons. Flat space maximizes entropy, while the insanely curved space near a microscopic black hole maximizes it. I also like this because if you get right down to it, entropy is really all about smoothness, in multiple forms. So: Is the inverse of space curvature considered an entropy metric? If not, why not? What am I missing?


title: Equations of motion for controlled/driven classical systems? Does D'Alembert's principle apply?

text: I'm puzzled about how to derive the equations of motion for certain classical systems where some entity is controlling some of the DOFs. For example, consider a double-pendulum, with lengths $l_1$ and $l_2$, masses $m_1$ and $m_2$, and which deviate from the vertical by $\theta_1$ and $\theta_2$. Let's say that at the joint between the two pendulums (pendula?), there is a little device controlling the angle $\theta_1+\pi-\theta_2$, and forces it to be some function of time: $\theta_1(t)+\pi-\theta_2(t)=g(t)$. If I understand correctly, this is a holonomic constraint, since it is simply a functional relationship between the DOFs and t. Obviously the position and velocities of the masses can then be expressed just in terms of $\theta_1(t)$, $g(t)$, and their derivatives. My question is: can we then determine the equation of motion for $\theta_1$ with the Euler-Lagrange equations in the "usual" way? The reason I have doubts here is that it seems like the Euler- Lagrange equations come from D'Alembert's principle, where the constraints do no "virtual work" on the system; and I'm not sure the constraint in this case satisfies this.


title: Secondary constraints leads to the value of lagrange multiplier

text: From Lagrangian I got two primary constraint $\phi_i$ and $\phi$. And my Hamiltonian in presence of the constraints becomes- $$H_p=p\dot q-L+\lambda_i\phi_i+\lambda\phi$$ here the $\lambda_i$ and $\lambda$ are Lagrange undetermined multiplier. Now from $\dot \phi_i=[\phi_i,H_p]$ I got secondary constraint $\Sigma_k$ and from $\dot \phi=[\phi,H_p]$ I got another secondary constraint $\Sigma$ . To satisfy the consistency condition I calculated the $\dot \Sigma_k=[\Sigma_k,H_p]$ and $\dot\Sigma=[\Sigma,H_p]$. $$$$ From the relation I have $\dot \Sigma\approx0$. But the $\dot \Sigma_k$ gives the value of of $\lambda_i$. Now can anyone help me how can I further analyze the constraints in this case? Do I have to put the value of $\lambda_i$ in the equation of $H_p$ and calculate the commutation again? An example would be lovely.


I am trying to find the following: **How many pieces of toast would you need to make a black hole?** From what I've learnt so far I need to find an equation for the compression force the massive amount of the intergalactic pile of pieces of toast would exert on itself to be able to compress it to within a Schwarzschild radius. As you can tell, this is not an easy or normal question. Any help is appreciated.


title: Why isn't there a potential difference across a disconnected diode?

text: I know this question sounds silly, as if there was a potential difference a current would be created when the terminals are connected together and this would mean energy has come from somewhere. The reason I ask this though is that from my understanding of the depletion region and built in potential of a diode it seems like if you connected a voltmeter across the whole diode it would show the value of the built in potential. This is explained in the image below: ![pn junction under equilibrium bias](http://i.stack.imgur.com/SaSH6.jpg) At first, electrons flow from the n type to the p type because there are a higher concentration in the n type, and holes do vise versa. This is called the diffusion current. The first electrons and holes to cross the pn boundary are the ones which are closest to it; these carriers recombine when they meet each other and are then no longer a carrier. This means there is a depletion region of no carriers near the pn boundary. because electrons have left the n type material, and holes have left the p type material, there is a surplus of positive and negative charge on the n and p side of the pn boundary respectively. This causes an electric field that opposes the diffusion current, and so no more electrons or holes cross the boundary and combine. In short, only the electrons and holes near the boundary combine, because after they have done that an electric field is formed that prevents any more carriers from crossing. The current due to this electric field is called drift current, and when in equilibrium this will equal the diffusion current. Because there is an electric field at the boundary (pointing from the positive charge to the negative charge) there is an associated voltage. This is called the built in potential. If you sample the electric field at each point along the diode from left to right, you would start with 0 in the p region because there are an equal number of protons and electrons. As you approach the depletion region you would see a small electric field pointing back towards the p region, caused by acceptor impurities which now have an extra electron (due to recombination) and therefore now have a net negative charge. This electric field would increase in strength as you get closer to the boundary, and then die away as you get further away. This electric field means there is a voltage, as shown in graph (d). The p side is at an arbitrary potential, and the n side is at a potential higher than this because there is an electric field between them. This means there is a potential difference across the depletion region; this is known as the built-in potential. But why when I connect a volt-meter across the whole diode will I not see this built in potential? **Edit:** I've also asked this question on another SE site And they say that the reason is because the net potential difference across the diode once it has been connected to a wire or volt meter is zero, because a metal-semiconductor junction has an electric field between it, and the overall effect of the electric fields at the junctions of the m-p-n-m materials results in zero potential difference. Regardless of the fact that voltage does not equal electrostatic potential difference, can you please confirm that there is also no net electrostatic potential difference across a diode connected to a wire or volt meter because of the m-s junctions? I'd like to get a physicists answer on this, to see if this is the truth or just a simplified explanation to make it easier to accept.


title: What kind of problem in quantum mechanics can have an algebraic method of solution?

text: For example, a harmonic oscillator can have an algebraic solution, and hydrogen potential can   also have an algebraic solution. Here the algebraic method of solution means that we can use the similar methods like $a$ and $a^\dagger$ to solve QM problem. So in general what kind of problem in QM can have an algebraic solution. For example, can infinite potential well have a algebraic solution, and so on.


title: Acceleration ratio

text: Can anyone please explain why: $$a_1=a_2\tan\alpha $$ ![enter image description here](http://i.stack.imgur.com/9IPBy.jpg) We have 2 **same** cubes placed on a platform, between them there is a wedge which is pushing the cubes in opposite directions. The hard part of the question is to find the ratio between the cubes   accelarion and the wedge acceleration, which is given above, and I cant understand why. I can't understand the physical idea behind this.


title: Lorentzian interpretation of open-closed string duality

text: In open-closed string duality, we can reinterpret the one-loop open string diagram (an annulus or cylinder) as a propagating closed string, depending on the direction in which we take time to be. This duality is reflected in the amplitudes we attach to these two different interpretations of the cylinder diagram. My question concerns the interpretation in a target space with Lorentzian signature: surely whether we treat the string worldsheet as a loop of open string or a propagating closed string depends on the reference frame? It could for example be that in one reference frame, surfaces of target space simultaneity intersect the cylinder as (two virtual) open strings, but when we boost to another reference frame, the surfaces of simultaneity intersect the cylinder as a closed string. Is this possible and, if so, what would it mean, if anything?


title: What is a Zero-Phonon Line (ZPL)?

text: I am trying to understand the electronic structure of the negatively charged NV centre in diamond, where there is a so-called Zero-Phonon Line (ZPL) in the spectrum. Can anybody explain what a ZPL is?


title: Why do white dwarfs shine white?

text: The hotter something is glowing the more white/blue it appears. A dying medium sized star expands, cools and becomes a red giant for a while, but eventually it is going to gravitationally collapse (once enough Iron (Fe) is accumulated in the core). Then it blows the outer layers away and what is left collapses into a white dwarf. What makes the dwarf shine? and why is it white? Does the luminosity decreases as the object cools down, or is there some other reaction that keeps it glowing for a long time? Can a white dwarf turn brown or black never to be seen again? Do all white dwarfs turn into Neutron stars eventually?


title: Matter in superconductive state

text: We distinguish between the states of matter: gas, liquid and solid. Possibly we could add the plasma state and/or the superconductive state as new states of matter. Phase transistions at certain temperature perhaps with some other conditions should have to exist. What do you think, does it make sense?


title: Boson calculus and Maximum Weight State

text: I'm just going over a few past exams for tomorrow, and I've come across a question that I'm having quite a bit of difficulty with. > Let $\left|0\right\rangle$ denote the Fock vacuum state so that $b_j > \left|0\right\rangle = 0$, for all $j$. For any positive integer $N$, show > that the state $(b_1^{\dagger})^N \left|0\right\rangle$ is a maximal weight > state of $gl(3)$ formed by $a_{jk} = b_j^{\dagger} b_k$ Conceptually, I'm just kinda unsure what I'm meant to be doing. Any help would be awesome. :)


title: What happens to entangled particles when momentum is measured?

text: In Wikipedia it is mentioned that position and momentum can be entangled as well as spin and polarization etc. I assume etc. is charge etc. I understand how if you measure spin up on one of a pair you get spin down on the second of the pair. What happens to the other particle in an entangled pair if I measure the momentum, position or charge of one of the particles? Is there a momentum up and down or charge up and down analog? http://en.wikipedia.org/wiki/Quantum_entanglement


title: Can physics (ever) explain intrinsic properties of nature?

text: I may be totally off with this quite abstract (?) question(s). But still, here are some closely related sub-questions:   * Is there a list of currently "known" intrinsic properties of nature?   * How exactly is an intrinsic property of nature defined? Is it defined as " _it is so, because it is so_ " or " _it is so, because it cannot be otherwise (in our Universe)_ "? For example, Wikipedia about mass as intrinsic property: > _An intrinsic property is a property that an object or a thing has of > itself, independently of other things, including its context. An extrinsic > (or relational) property is a property that depends on a thing's > relationship with other things. For example, **mass is an intrinsic property > of any physical object**._ Is this a correct explanation of an intrinsic property?


title: Is there any antenna with a single null?

text: If we designate the origin (the reference point from which all displacement vectors are measured) $\vec{0}$, and If we consider a sphere $\mathbb{B}\left(\vec{0},\mathcal{R}\right)$ of radius $\mathcal{R}$ and centered at $\vec{0}$, and say that outside this sphere the charge density $\rho$ and current density $\vec{J}$ are zero at all points and all instants, then it can easily be shown pretty directly using Jefimenko's Equations that for points where $r \gg \mathcal{R}$, if we define: $$ \vec{\mathcal{Q}}\left(\hat{r}, t\right) = \iiint_{\mathbb{B}\left(\vec{0},\mathcal{R}\right)} \frac {\partial} {\partial t} \vec{J} \left(\vec{s}, t+\frac {\vec{s}\cdot\hat{r}} c\right)\space dV\left(\vec{s}\right) $$ Then we have $$ \vec{E}\left(\vec{r},t\right) \approx \frac {\mu_0}{4\pi r}\left(\vec{\mathcal{Q}}\left(\hat{r}, t-\frac{r}{c}\right)\times\hat{r}\right)\times\hat{r} $$ $$ \vec{B}\left(\vec{r},t\right) \approx \frac {\mu_0}{4\pi rc}\left(\vec{\mathcal{Q}}\left(\hat{r}, t-\frac{r}{c}\right)\times\hat{r}\right) $$ $$ \vec{S}\left(\vec{r},t\right) \approx \frac {\mu_0}{16\pi^2 r^2c}\left|\vec{\mathcal{Q}}\left(\hat{r}, t-\frac{r}{c}\right)\times\hat{r}\right|^2 \hat{r} $$ Now we can apply the Hairy Ball Theorem to deduce that the vector $\vec{\mathcal{Q}}\left(\hat{r}, t-\frac{r}{c}\right)\times\hat{r}$ must be zero for at least one value or $\hat{r}$, which as we know rules out the isotropic antenna. Now my question is, since it is allowable to have exactly one direction $\hat{r}$ in which this vector (and consequently, the radiated power) is zero, **is there any such antenna** \--one that exhibits a **non-zero radiated power in all directions** $\left(\theta,\phi\right)$ **except one**? Somewhat like the pattern exhibited by a Cardioid microphone \-- a cardioid revolved around its axis?


title: Force on bolt holding up jet engine on plane

text: I am to solve the following: > A jet engine of mass $m$ is fastened to the fuselage of a passenger jet by a > bolt. During flight, the plane encounters turbulence, which suddenly imparts > an upward vertical acceleration of $2.60\mathrm{~m/s^2}$ to the plane. > Calculate the force on the bolt. Since the jet engine weighs $9.81m\mathrm{~N}$ and the engine has a net force of $2.60m\mathrm{~N}$, the force on the bolt is either $(9.81+2.60)m\mathrm{~N}$ or $(9.81-2.60)m\mathrm{~N}$, but I do not understand which. I would like to draw a free-body diagram to help me understand the problem, but I am not sure how to in this case. **UPDATE:** Here is what I have. Suppose the mass of the fuselage is $M$ and the mass of the bolt is negligible. Then the free-body diagram is: ![free-body-diagram](http://i.stack.imgur.com/hHaBi.png) where   * $F_{FA}=2.60M$ is the force on the fuselage by the air   * $F_{AF}=Mg$ is the force on the air by the fuselage   * $F_{BE}$ is the force on bolt by the engine   * $F_{EA}=2.60m$ is the force on the engine by the air   * $F_{AE}=mg$ is the force on the air by the engine However, I am not sure if this is correct.


title: What do the line segments on the BICEP2 B-mode polarization map mean?

text: ![BICEP2 B-mode Signal](http://i.stack.imgur.com/eHV8x.png) The first image of BICEP2 visuals shows the "BICEP2 B-mode Signal", described as follows: > Gravitational waves from inflation generate a faint but distinctive twisting > pattern in the polarization of the CMB, known as a "curl" or B-mode pattern. > For the density fluctuations that generate most of the polarization of the > CMB, this part of the primordial pattern is exactly zero. Shown here is the > actual B-mode pattern observed with the BICEP2 telescope, with the line > segments showing the polarization from different spots on the sky. The red > and blue shading shows the degree of clockwise and anti-clockwise twisting > of this B-mode pattern. I think I understand the red and blue, but I don't get what the line segments mean. Another web page explains a similar visual as follows: > Summing over all incident waves [at a given point], the E-fields are roughly > equal in all directions, but not quite. There will be one direction that has > a slightly greater magnitude of E than the other directions (see figure to > the left). > > We can represent polarization as a line with length proportional to the > excess magnitude in that direction and at an angle such that it is aligned > with the direction of largest E. Aside from the fact that the latter description refers to E-mode, seemingly treating it as isomorphic to B-mode polarization (I'm talking through my hat a bit here)... I would expect from my understanding of this description that the length of a line segment should correlate to (the "absolute value" of) the intensity of red or blue at that point. But in the first visual referred to above, the correlation doesn't really seem to hold. (BTW, I'm assuming that each line segment represents information about the point at the _center_ of the line segment, not about a point at one end, unlike a vector field visualization would. That's because these are not vectors, since they don't really have direction in a 360-degree sense, but rather they have 180-degree rotational symmetry. This is consistent with the fact that each line segment's _center_ seems to be on a grid point. Pardon me if I'm belaboring the obvious, but it took me a while to realize this.) So... what am I misunderstanding here? What aspect of the polarization do the line segments actually show? To summarize the question a different way: what is the difference between what the length of the line segments represents, and what the intensity of the red/blue color represents?


The observed Higgs boson mass is at an interesting place in parameter space, placing the standard model electroweak vacuum right at the edge of metastability. Among the proposed explanations of this value is the existence of a "shift symmetry" in the Higgs sector. Today we have the possible detection of gravitational waves produced during inflation, of an amplitude suggesting that the inflationary potential was flat right up to near the Planck scale. Various expert commentaries (McAllister, Reece) say that one would expect interactions between the inflaton and the ultraheavy degrees of freedom to appear, and their absence might, once again, imply the existence of a shift symmetry. One of the many many approaches to inflation is "Higgs inflation": the Higgs field also serves as the inflaton field, the source of inflation. In discussions of Higgs inflation, I've been told that it's an unlikely model for exactly the same reason as mentioned above - effective field theory ought to break down near the Planck scale, the flatness of the potential should be disturbed by ultraheavy interactions, Higgs inflation would require finetuning. But now we may have evidence of such finetuning in inflation! Or perhaps, evidence of a protective symmetry. So my question is, **Could the same symmetry be finetuning both the Higgs mass and the inflaton's interactions?** Could the same shift symmetry make the Higgs mass critical _and_ protect the inflaton from ultraheavy interactions? These questions might be asked first in the context of basic Higgs inflation - only one fundamental scalar - and later in the context of a multi-scalar theory, in which the Higgs is part of a larger scalar sector with a single big potential. **edit** : I have found a discussion of shift symmetry in the context of Higgs inflation, but it was written prior to the measurement of the Higgs mass.


title: Does light change color on its way through a window?

text: Looking at the refractive index of glass, it's around $1.6$. Then the speed of light $x$ through light should be given by $$ 1.6 = \frac{3.0\times10^8}{x}, $$ so $x$ is about $2\times10^8~\mathrm{m}~\mathrm{s}^{-1}$ The frequency is kept constant, so the wavelength must adapt to suit the slower speed, giving a wavelength of $2/3$ the original. Does this mean that when passing through glass, say red light (wavelength $650~\mathrm{nm}$) changes to indigo ($445~\mathrm{nm}$), as $650 \times 2/3 = 433~\mathrm{nm}$, or is my logic flawed somewhere?


title: What determines color -- wavelength or frequency?

text: What determines the color of light -- is it the wavelength of the light or the frequency? (i.e. If you put light through a medium other than air, in order to keep its color the same, which one would you need to keep constant: the wavelength or the frequency?)


title: What made the keys in my pocket so hot?

text: Not sure where to ask this question - thought you guys would probably have the best idea! Today a single key on my keychain in my pocket heated up so that it was too hot to handle and scalded my leg. Any ideas what could possibly have done this? I always keep a prison guard style set of keys attached to my trousers and in my pocket - one of these keys is very large and thick (longer and thicker than my finger) - this is the key that heated up somehow. In my pocket was my phone (an htc desire hd). I was sitting at my desk as I do all the working day - when I felt my pocket getting hotter rather alarmingly quickly. I stood up and pulled everything out of my pocket - and I can't work out how it happened. The phone was not hot at all and worked fine. My best guess is that there was some form of induction going on. Does anybody have any idea what might have happened? I don't think it was friction or bending of the keys as it's never happened before and I was sitting still all morning. Thanks for any ideas! I'm more curious than anything else.


title: matrix field theory

text: I am studying a field theory where the field is a matrix. The problem is that I have to calculate some functional derivative. How could we define functional derivative when the field is a matrix ?


title: Showing that an operator is Hermitian

text: Consider the operator $$T=pq^3+q^3p=-i\frac{d}{dq}q^3-iq^3\frac{d}{dq}$$ defined to act on the Hilbert Space $H=L^2(\mathbb{R},dq)$ with the common dense domain $S(\mathbb{R})$. Here $S(\mathbb{R})$ denotes Schwartz space. How would I show that this operator is hermitian? I understand the procedure of using the scalar products, i.e. $\forall f,g\in S(\mathbb{R})$ $$<f,Tg>=<T^*f,g>=<Tf,g>$$ But, I am unsure if I could just apply complex conjuagtion right to $T$ to test the operator. It seems if I do this that the operator is not hermitian as the minus sign switches to positive. This leads me to believe this method is flawed and a valid test of the operator being hermitian. The generalization of this operator to $$T_n=pq^n+q^np$$ and its importance as an example of the subtle difference between hermitian and self-adjoint matrices defined on infinite dimensional Hilbert Spaces $H$ and usage with deficiency indices led me bring the question up.


title: Modeling the diffraction of arbitrary electric fields by diffraction gratings

text: I have read from many sources that the Fraunhoffer diffraction pattern is the Fourier transform of the grating. But I feel like these explanations do not take into account the curvature of the initial beam. Let's say my grating is defined by $G(x)$. I have a beam $E(x)$ incident on the grating and want to know what my image looks like in the far-field. The normal technique for modeling diffraction seems to be to just take a fourier transform: $\mathscr{F} (G(x))$. How do I interpret this in spatial variables? I can see that the answer makes sense in the Fourier domain, but how can I map this again onto the spatial domain without taking a trivial $\mathscr{F}^{-1}$ back to my original function? Also what if my initial beam has some arbitrary curvature. This method does not utilize any information about the input beam. I've thought of the following variations: $$ \mathscr{F}^{-1} \left[ \mathscr{F}\left[G(x)\right]\times \mathscr{F}\left[E(x)\right] \right]$$. and $$\mathscr{F}^{-1} \left[ \mathscr{F}\left[G(x) \times E(x)\right] \right]$$. I am having some numerical issues in visualizing the results so I can't confirm my answers. I'd appreciate any explanation of whether these variations are sensible or suggestions of what books might be useful. I've browsed through Goodman and a couple of other derivations of diffraction patterns but found nothing that seems to talk about arbitrary initial beams.


title: Use the relative velocity formula to find v2f in terms of v1f?

text: > **Q:** A $0.150\text{ kg}$ glider is moving to the right ($+x$) on a > frictionless, horizontal air track with a speed of $0.80\text{ m/s}$. It has > an elastic collision with a $0.300\text{ kg}$ glider moving to the left > ($-x$) with a speed of $2.20\text{ m/s}$. > > **a.)** What is the initial momentum of each glider? Express the momentum in > terms of unit vectors. > > **b.)** Use the relative velocity formula to find $v_{2f}$ in terms of > $v_{1f}$. > > **c.)** Use the relative velocity result to solve conservation of momentum > to find the velocity (magnitude and direction) of each glider after the > collision. I've figured part (a) using the definition of momentum: $p=mv$: **1st glider** : $$p_1 = m_1 v_1 = (0.15\text{ kg})(+0.8\hat{x}\text{ m/s}) = +0.12\hat{x}\text{ kg m/s}$$ **2nd glider** : $$p_2 = m_2 v_2 = (0.3\text{ kg})(-2.20\hat{x}\text{ m/s}) = -0.66\hat{x}\text{ kg m/s}$$ Parts (b) and (c) are what have me confused at the moment. I'm not positive I have the equations for relative velocity right nor how to solve for $v_{2f}$ in terms of $v_{1f}$. My book lists this an equation that can be gotten from manipulation of a kinetic energy equation: $v_{1i} – v_{2i} = -(v_{1f} - v_{2f})$. Is this the relative velocity formula? would just isolating $v_{2f}$ in this equation be solving for $v_{2f}$ in terms of $v_{1f}$? Any help appreciated. thanks!


title: Hamilton's equations in terms of initial conditions

text: I'm trying to understand the way that Hamilton's equations have been written in this paper. It looks very similar to the usual vector/matrix form of Hamilton's equations, but there is a difference. $$\frac{{\bf dZ}(t)}{dt} ~=~ J \frac{\partial H({\bf Z(t)})}{\partial z}, \qquad\qquad{\bf Z(0)} ~=~ z,$$ where $J$ is the block matrix $((0,1),(-1,0))$. $\bf{Z}(t)$ represents the point in phase space (positions and momenta). The part I don't understand is that the derivative of $H$ is taken with respect to the initial value (lowercase $z$) of ${\bf Z(t)}$. How does this form follow from the usual way of writing Hamilton's equations in vector form? Note that my question is not about the vector notation for Hamilton's equations, which can be found in any introductory textbook on classical mechanics. I am specifically asking about the derivatives with respect to $z$, the initial values of ${\bf Z(t)}$. Edit: I could not find an arxiv version of the paper, but it seems to be available as the first publication on one of the authors' websites, under the section "Coarse-graining with proper dynamics." The equation in question is the first one in section II of the paper.


title: Pn junction voltage drop?

text: This image from wikipedia, explains that there occurs a potential drop across a pn semiconductor junction, and an electric field confined to the depletion region.![enter image description here](http://i.stack.imgur.com/7VaN3.png) I already know the reason for the existence of this drop and the calculation of the difference but I have two questions regarding this drop. **1) If then and p doped regions are externally connected using a perfectly conducting wire, why will not any current flow?**   Connecting the two regions should equalize their potentials (since wire is a resistance less) conductor, and therefore the gradient at the junction is destroyed resulting in a diffusion current which is obviously against conservation of energy as the semiconductor has non-zero resistivity. Where will extra potential drops will be created so that Kirchoff's voltage rule holds without any current and the built-in potential difference $V_o$ of the juction persists? **2)** (probably naive) If an external Bias is applied, of say $|V|<|V_o|$, then the pn potential difference across the junction will just reduce by that amount ($|V|$). Assuming that the external voltage source is ideal without any resistance, **what will be the potential drops which would sum to zero in this case? Will drop due to the resistance of semiconductor play a par in it? If yes, then there cannot be any ideal resistance-less semiconductor junction, can there?**


title: Understanding operations of quantum computing advantages

text: For example, let us examine the case of quantum (discrete) fourier transform. There are $2^N$ samples. How do we initialize these $2^N$ samples into $N$ qubits? I have a hard time understanding this.


title: Some questions on the Wilson loop in the projective construction?

text: Based on the previous question and the comment in it, imagine two different mean-field Hamiltonians $H=\sum(\psi_i^\dagger\chi_{ij}\psi_j+H.c.)$ and $H'=\sum(\psi_i^\dagger\chi_{ij}'\psi_j+H.c.)$, we say that _$H$ and $H'$ are gauge equivalent if they have the same eigenvalues and the same projected eigenspaces_. And the Wilson loop $W(C)$ can be defined as the trace of matrix-product $P(C)$(see the notations here ). Now my questions are: (1)"$H$ and $H'$ are gauge equivalent" if and only if "$W(C)=W'(C)$ for all loops $C$ on the 2D lattice ". Is this true? How to prove or disprove it? (2)If the system is on a 2D torus, is $W(L)$ always a _positive real number_ ? Which means that the 'total flux'(the phase of $W(L)$) through the torus is quantized as $2\pi\times integer$, where $L$ is the boundary of the 2D lattice. (3)If the Hamiltonian contains extra terms, say $H=\sum(\psi_i^\dagger\chi_{ij}\psi_j+\psi_i^T\eta_{ij}\psi_j+H.c.+\psi_i^\dagger h_i\psi_i)$, is the Wilson loop still defined as $W(C)=tr(P(C))$? Thanks a lot.


title: Problem with Velocity of efflux

text: I am stuck in this problem- ![The Problem](http://i.stack.imgur.com/CBC4d.png) I need to find the velocity of efflux at the hole of the container. [We can assume that the area of the hole is negligible in comparison with the base area of the container]. Here's my approach Velocity of liquid at the upper-surface = $v_2$ Velocity of efflux (velocity of water at the hole, right?) = $v_1$ Using Bernoulli's equation for the surface and the hole - $$ P_{atm} + \rho_2 g (h_1 + h _2) + \frac{1}{2}\rho_2 v_2^2 = P_{atm} + \rho_1 g h_1 + \rho_2 g h_2 + \frac{1}{2}\rho_1 v_1^2 \\\ \implies \rho_2 g h_1 + \rho_2 g h_2 - \rho_1 g h_1 - \rho_2 g h_2 = \frac{1}{2}(\rho_1 v_1^2 - \rho_2 v_2^2) \\\ \implies \frac{1}{2}(\rho_1 v_1^2 - \rho_2 v_2^2) = g h_1 (\rho_2 - \rho_1) $$ Now, let area of the base be $A_2$ and that of the hole be $A_1$ then, using equation of continuity, $$ A_1 v_1 = A_2 v_2 \\\ \implies v_2 = \frac{A_1}{A_2} v_1 \\\ \implies v_2 \approx 0 (\because {A_1 << A_2}) $$ Using this value in the previous Bernoulli's relation $$ \frac{1}{2}(\rho_1 v_1^2) = g h_1 (\rho_2 - \rho_1) \\\ \implies \frac{1}{2} \rho_1 v_1^2 = g h_1 (\rho_2 - \rho_1) \\\ \implies v_1 = \sqrt {\frac {2gh_1(\rho_2 - \rho_1)}{\rho_1}} $$ Which is not the correct answer. I did get a correct answer in the chat room, but it was using a different method. What's wrong with my method?


title: If time travel to the past is not possible why is this situation considered in Feynman diagrams?

text: I recently read R.P Feynman's _QED:A Strange Theory of Light and Matter._ It is believed that time travel to the past is not possible. Then why is particles going backward in time considered in the book while drawing Feynman diagrams?


title: Why is the change of temperature $\Delta T$ measured in Kelvins, degrees Celsius, etc.?

text: Let me start by apologizing if this question seems pedantic and say that I'm not very familiar with physics in general, as I'm a math major instead. Anyway, say a body changes from temperature $T_1$ to $T_2$, with $T_2 \ge T_1$. Then the change in temperature is $$\Delta T = T_2 - T_1$$ Now, it's clear that if $\Delta T = x\text{K}$ then $\Delta T = x \text{°C}$, with $x \ge 0$. But it's also clear that $x \text{K} \ne x \text{°C}$, which leads to a contradiction. Then I don't really understand why are units typically written in $\Delta T$? I suppose it could be to illustrate the units used to measure $T_1$ and $T_2$, but is it really necessary?


title: How does linearity of a measurement imply that the commutator of all measured observables are $c$-numbers?

text: I really don't understand with the linearity conditions I have where this comes from.


title: Ramsey Interactions

text: What are Ramsey interactions? I am researching atomic clocks and am not sure why the atoms need to be exposed twice to an electromagnetic field in order to cause excitation.


title: What is "Symmetric Fission"?

text: Dose anyone has a clue what Symmetric Fission is? I couldn't find any explanation on what is it on internet.


title: Bose-Einstein condensate and nonlinear waves

text: Can Bose-Einstein condensate be written as non-linear wave equation (in terms of mean field approximation theory)? the equation is: ![enter image description here](http://i.stack.imgur.com/X8kEs.png) source: http://xxx.tau.ac.il/abs/1308.2288 > What I do understand by the Bose-Einstein condensate is, it is the state of > atoms at very low temperatues and at the moment the atoms forget their > previous identity. Therefore all the atoms stay in the same quantum state. Am I right?


title: future light cones and light paths

text: I understand that an event, in a four dimensional space-time, produces a light cone. As time increases the cones gets larger on either side of the event (past and future). For example the if the sun where to "go out" it would take 8 minuets for the earth to be affected by it simply because it takes approximately 8 minuets for light from the sun to reach the earth due to its location in the future light cone of the event (the sun being the event). Einstein made a suggestion that space-time is warped (vs flat) by the distribution of mass and energy and that bodies (like earth, jupiter, etc.) are meant to follow straight paths but cannot in a warped space (or appear not to because space is warped). However light supposedly follows these warped paths (called geodesics) as well even tho light is energy. How can space bend light? and why does light have to follow a specific path which is warped by space? Things with mass (like water) have to go around things (like rocks), but light can go through certain things or expand, but it doesn't move to the side like water does to a rock. Does it?


title: Which person can handle falls from big heights better: lighter or heavier?

text: This probably includes a little of biology, but I believe it's mainly physics, so I hope it's ok to ask here: Imagine 2 persons: Person A weights 120 pounds and person B weights 180 pounds. Imagine they both fall from a big height:   a) landing on their knees (a good jump)   b) falling on their back (a bad fall) For both of those persons, in both situations, for which of them the fall is easier to handle? I reckon:   1. when falling down, the speed is not relative to weight, therefore both those persons will hit the ground at the same speed   2. but, I assume, as the heavier person will create bigger force to the ground, the ground will hit him back with the same force, therefore a heavier person should get a bigger blow when hitting the ground   3. now, this one I'm not sure about, but I assume that a heavier person (considering it's not an unhealthy, fat person, but a healthy one, with muscles, etc) is able to withstand a greater force/blow Assuming the last assumption is correct, how are assumptions 2 and 3 related? Is there anything I missed? Overall, which person can withstand falls from greater heights: a lighter one, or a heavier one?


title: Gyrochronology, the formula is empirical ?

text: The formula used in Gyrochronology that relates a star's Period of Rotation- Mass-Age is empirical? This news How to Learn a Star’s True Age > "“A star’s rotation slows down steadily with time, like a top spinning on a > table, and can be used as a clock to determine its age," pointed to this paper THE KEPLER CLUSTER STUDY: STELLAR ROTATION IN NGC6811   and later I found this one (gyro_background) with original work. But I couldnt find a justification for the formula. Is the Period proportional to $age^{{1/2}}$ only an empirical result? It seems to me that the formula is a data fit and not a direct result of a calculation of the stellar intrinsic dynamics. The rate of mass loss by radiation must have a 'word to say' in the formula.   Any help is welcome.


title: Can the entropy density of a spacelike singularity arbitrarily exceed the inverse Planck volume?

text: For the purpose of this question, let's restrict ourselves to BKL singularities. BKL cosmologies are homogeneous Bianchi type XIII and IV cosmologies which exhibit oscillatory chaotic behavior, although that's not relevant to this question. Most generic singularities can be approximated locally by a BKL solution. The volume of a BKL universe decreases linearly with time as the singularity approaches. If the matter falling toward the singularity has a nonzero entropy, and the second law of thermodynamics is satisfied, the entropy density will increase without limit as the singularity approaches. Can the entropy density exceed the inverse Planck volume? Being inside a black hole, the holographic bound does not apply. If entropy densities beyond the inverse Planck volume are forbidden, is the second law violated?


title: Teflon and induction cookware

text: I remember once getting new Teflon (non-stick) cookware; however, when I tried this new cookware on my induction cooker the _cookware did not heat up_. My regular steel cookware worked just fine on the induction cooker, both before and after trying the Teflon cookware. I've heard from at least two other people that they've noticed the same thing with Teflon cookware and induction cookers (at least one tried the same exact brand and style of cookware as myself). And so, I wonder: Did my Teflon cookware not heat up with the induction cooker because of the Teflon or the process it went through to be Teflon coated? In other words, does Teflon cookware in general not work with induction cookers? Or is it possible that my Teflon cookware just happened to not be made of a ferrous material and thus the induction cooker had no effect on it, with or without the Teflon? If it is the case that Teflon in no way affects the performance of the cookware on an induction cooker, is it possible that a majority (or all) of Teflon cookware is made of non-ferrous material (perhaps because it is easier to get the Teflon to stick to aluminum than is to stick to steel)? In this way Teflon cookware would not work on induction cookers but not because of the Teflon. I unfortunately no longer live in the same apartment as the induction cooker or the Teflon cookware so I cannot verify whether the cookware is ferrous or not myself.


title: What is different in representation?

text: I'm sorry if this is somewhat a dumb question. First: "Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as _linear transformations of vector spaces_ " I know little about particle physics, but to what I know, physicists only deal with the groups of (linear) symmetric operators acting on vector space of states. So in fact physicists are dealing with the _italic_ part of the representation theory. Why would them bring it in? What is a significance of the action "representing an element of a group as linear transformation" in the work of physicists, who are already dealing with groups of linear transformations?


title: $B\mu$ from $\tan\beta$ and $\mu$

text: I'm using the SOFTSUSY package to generate the sparticle spectrum at the EW scale. One of the input parameters is the ratio of the up and down-type Higgs vevs commonly known as $\tan\beta$. The $\mu$ parameter is computed as an output by constraining the Z-boson mass to be compatible with experiment. I was wondering if there a quick way to figure out the value of the $B\mu$ parameter from $\mu$ and $\tan\beta$. Even an approximate analytic expression would be nice to have.


title: Electromagnetic radiation and black body radiation

text: I was taught today that the Electromagnetic wave Theory is unable to explain black body radiation. The example that was given to me: When a metal is heated, it emits different frequencies of light as it gets hotter. If electromagnetic wave theory was correct, it would not be so, the frequency (color) of light would remain the same, but only the intensity will change. I don't understand why this is so. My logic: Electromagnetic waves occur when a charged body oscillates in a electric and magnetic field. If the metal is provided with more energy (in the form of heat) won't the charged body vibrate faster, thus changing the frequency of the light emitted?


title: Proof of the conservation of the energy functional for the Gross-Pitaevskii equation?

text: From the Gross-Pitaevskii equation \begin{equation}i\hbar\frac{\partial\psi}{\partial t}=\left(-\frac{\hbar^2}{2m}\nabla^2+V+g|\psi|^2\right)\psi\end{equation} using the variational relation \begin{equation}i\hbar\frac{\partial\psi}{\partial t}=\frac{\partial\varepsilon}{\partial \psi^*}\end{equation} we find the energy density \begin{equation}\varepsilon=\frac{\hbar^2}{2m}|\nabla\psi|^2+V|\psi|^2+\frac{g}{2}|\psi|^4\end{equation} The energy would be $E=\int d^3r \varepsilon$ and this is a prime integral of the motion, meaning it is a conserved quantity. My questions are: 1) How do we get the variational relation? 2)How can we prove that $E$ is a conserved quantity?


title: What's the critical temperature of the XY model on a triangular lattice

text: I've been looking deeply into many bibliographic references without finding the answer. I would be interested in knowing the numerical value of the critical 2d XY spin model on triangular lattice. Being derived analytically (if possible) or from Monte-Carlo doesn't matter.


title: Kinematics textbook illustration

text: ![enter image description here](http://i.stack.imgur.com/jKBIc.png) I have trouble interpreting this illustration. I see why **r** (position) and **a** (acceleration) are the way they are, but what happened to **v**? Why is it smaller than its coordinates? Is this another error in my textbook?


title: Where does the energy from a nuclear bomb come from?

text: I'll break this down to two related questions: With a fission bomb, Uranium or Plutonium atoms are split by a high energy neutron, thus releasing energy (and more neutrons). Where does the energy come from? Most books I've ever come across simply state e=mc2 and leave it at that. Is it that matter (say a proton or neutron) is actually converted into energy? That would obviously affect the elements produced by the fission since there would be less matter at the end. It would also indicate exactly how energy could be released per atom given the fixed weight of a subatomic particle. I remember hearing once that the energy released is actually the binding energy that previously held the nucleus together. With a fusion bomb, two hydrogen isotopes are pushed together to form helium and release energy - same question: where does this energy come from? Is matter actually converted or are we talking about something else here? Sorry is this is rather basic - I haven't done physics since high school.


title: What is meant by Vacuum?

text: What is the exact meaning of the word vacuum? Is it just a state of very low pressure or is it nothingness (as in there is nothing)? Also, when we say space is vacuum - it must be referring to pressure as space has light travelling (which means photons) besides the big masses of comets, planets, stars.


title: Thermal radiation in the Unruh Effect

text: The following formula has been given in 't Hooft's black holes notes ($|\Omega \rangle$ is the vacuum state of Minkowski space, O is a operator): $$\langle \Omega| O|\Omega \rangle = \sum_{n \ge 0} \langle n | O | n \rangle e^{-2 \pi n \omega}(1-e^{-2 \pi \omega})=Tr(O \rho_{\Omega})$$ **How does this mean that the radiation is thermal and follows Plancks black body law?**   1. Here, I read that $\langle O \rangle = \frac{1}{Z}\sum_n e^{-\beta E_n}\langle n |O|n\rangle$. How is this sum the same as that in the above expression?   2. How is the density matrix related to the law of black body radiation? How can I derive Planck's law from the expectation value of O in the first expression?


What is the difference and similarities between Stueckelberg mechanism and Higgs mechanism? They both make the gauge field massive. Is the Stueckelberg mechanism a special case about U(1) gauge fields of Higgs mechanism? Does there exist Spontaneously symmetry breaking in Stueckelberg mechanism?


title: Barbells and gravity

text: A giant set of bar bells floating in space (like two identical sized planets connected by a long rod) would have a centre of mass midway between the two on the connecting rod. But surely it would have two centres of gravity, one at each end? If you were standing on one of the "bells" or planets, and threw a rock in the air, it wouldn't fly to the middle of the rod, surely? And if I'm correct, then say, a big wobbly jelly shaped planet would also have multiple points of gravity. We have to except a sphere on which the centre of mass and gravity are the same. My interest in dark matter was brought about by a friend who explained that the observed mass was calculated with reference to the centres of galaxies - but in what sense can a galaxy have a centre if the above confusions come into play? Isn't a galaxy like a set of interconnected barbells? Is there really a "centre" for gravitational calcuations?


title: Analog of Klein-Gordon equation from Proca action

text: What would be the general form of Lagrange Equation when instead of a scalar field we have a vector potential? has anyone derived the klein gordon equation for a corresponding vector potential Lagrangian?


title: What's the difference between average absolute error and relative error?

text: I am quite confused by both these terms. I would like to know what's the exact difference between both these terms and which one is more accurate.


title: Is Gravity a part of the Strong Nuclear Force?

text: In episode 7 season 5 of Through the Wormhole, they say that gravity could just be a manifestation of the strong nuclear force and that the force carrier are a pair of Gluons. Is this valid in anyway?


title: What is the direction of the friction force on a rolling ball?

text: Suppose you have a solid ball on a horizontal table.   1. What is the direction of friction force when the ball I pushed horizontally and **starts rolling**?   2. Why is the direction of friction as it is?    3. Which forces acts at the contact point between delta time t0 to t1? (If we divide friction force in sub forces) V=1Vx m/s Fx=?


title: Newton's second law - infinite velocity?

text: Given the equation $$F_{\text{net}} = ma$$ Does this not imply that if the net force on a certain object is positive, its acceleration will also be positive, and theoretically this object would accelerate forever to an infinite velocity? i.e envision a block on a surface. If a person were to apply a force (push) on this block that exceeds friction, ad continuously applies this force, this block should technically accelerate infinitely?


title: How would I calculate celestial body characteristics without a priori knowledge?

text: Without any a priori knowledge of the mass, speed, distance, and size of local celestial bodies (aside from Earth's size), what can I calculate and how from my "backyard" through observation? ### Edit: My goal is to create a high-school level physics project that students can conduct over the course of a year/term as both a physics problem mixed with a bit of astrophysics history. Core concepts would include observation (within reason). This isn't meant to be extremely precise work (within an order of magnitude would be excellent), but to demonstrate the methods. Given a bunch of knowns, it's trivial to find out one unknown, but I am unsure of how to start from square one like the early physicists and astronomers did in times past.


title: Are diffeomorphisms a proper subgroup of conformal transformations?

text: The title sums it pretty much. Are all diffeomorphism transformations also conformal transformations? If the answer is that they are not, what are called the set of diffeomorphisms that are not conformal? General Relativity is invariant under diffeomorphisms, but it certainly is not invariant under conformal transformations, if conformal transformations where a subgroup of diff, you would have a contradiction. Or I am overlooking something important?


title: Examples of singularities in classical physics

text: I am a math teacher and I have to teach a topic called "Bruchterme" and "Bruchgleichungen" in german (I don't know the english word for it). For example $$ \frac{x^2 - 3}{(x - 2)x^2} + \frac{4}{x} + 2 $$ is a "Bruchterm" and $$ \frac{4x}{2x -3} = 4 - \frac{2x}{x-1} $$ is a "Bruchgleichung". Students have to learn, how to determine for which $x$ the term or equation is defined (i.e. the "singularities") and how to solve such equations. Now my problem is that most textbooks about this offer little to no interesting applications of this, especially no applications of determining for which $x$ the term is defined. Now I am looking for interesting examples from classical physics or engineering for this type of problem. Especially examples where singularities occur and are physically "interesting" in some way. Though it is for high school level, I don't want to restrict the question to trivial examples, but to classical physics. The only physical applications I have in mind are the following: Two parallel resistors $\frac{1}{R} = \frac{1}{R_1} + \frac{1}{R_2}$   Same type of equation: $\frac{1}{f} = \frac{1}{b} + \frac{1}{g}$ for the thin lens equation (where $b$ and $g$ are the distances from the image to the lens resp. from the object to the lense) However here the denominators are rather trivial and I don't see why it would be interesting to determine for which value for example of $b$ (if the other values are fixed) the equation is not defined. Gravitational force ($\propto \frac{1}{r^2}$) here one might discuss that the force goes to infinity for $r \to 0$. However the point of view of the curriculum is not to discuss such limits but just to look if one is allowed to plug in certain values for the variable or not (in a sense that dividing by 0 is not allowed or not defined without connection to limits).   1. Are there some interesting examples from physics where students see that it is worth to learn how to solve such equations (from type and perhaps complexity as in my example above)   2. Are there examples from physics where students see that it is worth to learn to determine for which values the term or equation is defined? **Edit (too long for a comment)** @Danu: I am teaching in the less known part of the german education system, called "Berufliche Schulen" more specifically I am at a school which focus is on technology, engineering and science. The system is a bit complicated with classes on very different levels. To make it short I have the motivation problem described above in a low level class and in a high level class (the level of the second one is equivalent to a German "Gymnasium" or even a bit higher in mathematics and physics) but the students are generally very interested in technology, engineering, physics and computer science (most of them want to take up a profession in this fields. Ranging from electricians or lab technician in the lower level classes up to enegineers or physicists in the higher level classes). Students are from 15 to 18 years old. Even in the lower level classes they are very interested in nontrivial physics even if they cannot grasp it conceptually or mathematically. So for motivation purposes some nontrivial, but for the students interesting examples would be good.


title: Representing a polarization vector for light as a 'manifold of two state'

text: **Explain me these projections please** **Context:** I was reading a paper (Phys. Rev. A 68, 052307) which involved **mesoscopic coherent states of light**. There, in order to calculate the uncertainty of a single **Poincaré angle** (see polarization parametrization, for both polarization elipse and Poincaré sphere) http://en.wikipedia.org/wiki/Polarization_%28waves%29#Parameterization) due to _shot noise_ (by calculating $|\langle \Psi_k|\Psi_a\rangle|^2$ where $\Psi_k$ and $\Psi_a$ are two mesoscopic coherent pulses with the same amplitude and on a same great circle on the poincaré sphere) I got found the following representation presented as a "manifold of two state in cartesian $(x,y)$ at $45^o$ from horizontal", where $(\Theta_k, \Phi_k)$ are Poincaré angles and $\gamma, \delta$ are **projections** on axes $x,y$, and $\alpha$ is the coherent amplitude: $|\Psi(\Theta_k, \Phi_k)\rangle=|\alpha \gamma(\Theta_k, \Phi_k) \rangle \otimes |\alpha \delta(\Theta_k, \Phi_k) \rangle$ $\gamma = (1-i)e^{i \frac{\Phi_k}{2}} cos\frac{\Theta}{2} +(1+i)e^{i \frac{-\Phi_k}{2}}sin\frac{\Theta}{2} $ $\delta = (1+i)e^{i \frac{\Phi_k}{2}} cos\frac{\Theta}{2} +(1-i)e^{i \frac{-\Phi_k}{2}}sin\frac{\Theta}{2} $ And I don't understand. Thinking 'classicaly', writing with the polarization elipse (parameters $\Psi$ and $\chi$) I got $\vec{E(t)}=|E|e^{i\omega t}[cos\chi,\pm i sin\chi]R_\Psi$ Where $R_\Psi$ is a rotation of $\Psi$ and the $\pm i$ account for clockwise and counter clockwise, but its different. I've put this here just for a base of what I expect the answer to cover. I've had many ideas of 'mappings' but none of them were good.


title: Thought experiment: Tethered galaxies - to the extreme

text: If two rocks were tied together with a tight, absurdly long, non-elastic rope, and placed on planets at either end of Earth's observable universe - or beyond - What would happen?   1. Is the structural integrity of the rope enough to "overcome" the expansion of the universe all along the rope, causing the rope to break, or the rocks to "slide off" the planets? Or will the rope expand?   2. If the rocks slide off the planets, and the planets were far enough apart, why would the rope not be able to move away from either planet faster than the speed of light?


title: Induced EMF in axle of car

text: A car moves on a plane road in east-west direction.At what latitude of earth should it move so that induced EMF in the axle connecting it's wheels is maximum? A. At the poles B. At the equator C. At a latitude inclined at 45 degree to the equator


title: A whole lot of doubts on Lorentz representation

text: Can someone tell me in layman's language how the $(1/2,1/2)$ represents a vector field and $(0,1/2)$ or $(1/2,0)$ represents spinors and $(0,0)$ represents scalar field. Please don't be pedantic on mathematics part. I didn't take a course on group theory yet. Give me physical arguments why this is true? I have come across this in QFT course that I am currently enrolled in.


title: Why does the co-efficent of kinetic friction of a block on an inclined plane not change when you increase the angle of the plane?

text: The coefficient of kinetic friction can be calculated using $\mu_k = F_k/F_n$. A change in angle does not affect the normal force, but doesn't a change in angle affect the friction? As in if the angle is steeper, wouldn't the object have a greater propensity to move down the slope, thus there will be less friction?


title: How air humidity affects how much time is needed for heating the air?

text: In cold weathers it is suggested to put a humidifier since the air gets too dry. I wonder how the humidity affects how much time is needed to get the air at a temperature of 20 Celsius degrees? I mean suppose you have a cold room and you want to heat the air, Will the process be slower or faster in relation to variations in humidity? Since the water tends to keep its temperature I feel the process will be slower with higher humidity, but I am not sure.


title: How does a knife cut things at the atomic level?

text: As the title says. It is common sense that sharp things cut, but how do they work at the atomical level?


title: Water Pressure Calculation for a volume of water at a given height

text: I'm looking to create a rain harvesting system. I have a 275 gallon IBC tote that is 48" x 40" x 46". I have an adapter for a 3/4 garden hose at the bottom of the IBC tote. I'm trying to figure out three things:   1. What is the pressure at the bottom of the tote, assuming that the tote is full?   2. Would the pressure from the column of water in the tote be able to reach an 8' tall planter 40' away via the hose?   3. How much would the pressure increase per foot that I elevated the tank?


title: Why should an area vector point normal to the surface?

text: Why is it that the direction of an area vector should be always along the normal drawn to the surface? Can't it also be some other angles with the plane?


title: Dark matter a medium for light propagation

text: Is dark matter a candidate to fill void left by luminiferous ether as a medium for light travel?


title: Why is velocity of outermost point on a rotating wheel double the velocity of centre of mass?

text: 'In the answers to one of the questions based on rotation of a disc in my physics book the answer includes the statement 'As we know that the velocity of outermost point on a rotating disc is double the velocity of center of mass'. But, I didn't know that and why is it like that? My thinking: I know that as we go away from the center of mass the tangential velocity of particles of the disc increases according to $v=wr$. But, how does that result in the above result?


title: Relationship between two eigenfunctions of the time-independent Schrödinger Equation in one dimension?

text: What is the relationship between two eigenfunctions of the time-independent Schrödinger Equation (in one spatial dimension) if they both have the same eigenvalue?


title: Local guage symmety implies causality

text: I read in a QFT book that local gauge symmetry implies causality. Could someone please explain that statement and why it's true? Thank you.


title: Matrix elements of a one-fermion operator (first and second quantizations)

text: I'm currently struggling with the expression of operators in second quantization. I did an exercise in which I had to consider a fermion in a central potential $V(\vec{r})$ and show that the matrix elements of $V(\vec{r})$ and $V_{ij} a_i a^{\dagger}_j$ were identical. I think I get this right. Yet, when I try to do the same with two fermions in a central potential (not considering any interaction between them), I end up completely lost. On the one hand (first quantization), I have \begin{equation} <a b ∣ V(r) ∣ cd > = \frac{1}{2} \int \int d^3 r_1 d^3 r_2 \Big(\phi_a^{\ast}(r_1)\phi_b^{\ast}(r_2) - \phi_a^{\ast}(r_2) \phi_b^{\ast}(r_1) \Big) [V(r_1) + V(r_2)] \Big(\phi_c^{\ast}(r_1) \phi_d^{\ast}(r_2) - \phi_c^{\ast}(r_2) \phi_d^{\ast}(r_1)\Big) \end{equation} (I should be able to further develop and simplify this expression, but I'm stuck with eight terms with no simplification on the horizon) and on the other hand (second quantization), I have \begin{equation}<a b ∣ \sum_{ij} V_{ij} a_i a_j^{\dagger} ∣ cd > = < ab ∣ V_{ac} a_a a_c^{\dagger} + V_{ad} a_a a_d^{\dagger} + V_{bc} a_b a_c^{\dagger} + V_{bd} a_b a_d^{\dagger} ∣ cd>\end{equation} I tried applying the anti- commutation relationship to this expression so as to simplify it, but unsuccessfully so far. I am more or less sure I'm missing something terribly obvious and I should feel ashamed about it. I would gladly appreciate some fresh insight on this problem.


title: Does the 4/3 problem of classical electromagnetism remain in quantum mechanics?

text: In Volume II Chapter 28 of the Feymann Lectures on Physics, Feynman discusses the infamous 4/3 problem of classical electromagnetism. Suppose you have a charged particle of radius $a$ and charge $q$ (uniformly distributed on the surface). If you integrate the energy density of the electromagnetic field over all space outside the particle, you'll get the total electromagnetic energy, which is an expression proportional to $c^2$. The energy divided by $c^2$ is what we usually call the mass, so if we calculate the "electromagnetic mass" in this manner we'll get $m = \frac{1}{2}\frac{1}{4\pi\epsilon_0}\frac{q^2}{ac^2}$. If, on the other hand, you took the momentum density of the electromagnetic field and integrated it over all space outside the particle, you'd get the total electromagnetic momentum, which turns out (for $v<<c$) to be proportional to the velocity of the particle. The constant of proportionality of momentum and velocity is what we call mass, so if we calculated the electromagnetic mass in this way we would get $m = \frac{2}{3}\frac{1}{4\pi\epsilon_0}\frac{q^2}{ac^2}$, which is $\frac{4}{3}$ times the value we got before! That is the 4/3 problem. Feynman claims that this fundamental issue remains when we move to quantum electrodynamics. Was he right, and if so has the situation changed since the 1960's when he was writing? I've seen claims on the Internet (I don't have the links) that the 4/3 problem is still there in QED, but instead of $\frac{4}{3}$ the coefficient is something closer to 1. Is that true, and if so what's the coefficient? All of this is of course related to issues of self- energy and renormalization. Any help would be greatly appreciated.


title: Calculating Length Scales from Passive Scalar Field

text: I have a set of PLIF images of a passive scalar advected in a turbulent flow. I'm wondering if it's possible to estimate the integral length-scale based on the images of the passive scalar, and if so, how would I go about it. Any references to journal articles about this are welcome. It seems that a 2D cross-correlation of the images would help, with defined peaks at various length-scales, but then you'd have to filter out a lot of noise.


title: Is crystal momentum really momentum?

text: Almost every solid state physics textbook says crystal momentum is not really physical momentum. For example, phonons always carry crystal momentum but they do not cause a translation of the sample at all. However, I learned that in indirect-band-gap semiconductors, we need phonons to provide the crystal momentum transfer to make happen electron transitions between the top of the valance band and the bottom of the conduction band. Along with absorbing or emitting photons, of course. Photons do carry physical momentum. For the purpose of momentum conservation, it seems that phonons do carry physical momentum as well. How can we explain this? ======================================== To put it more specifically, I drew a graph to tell the story: ![](http://i.stack.imgur.com/syVah.png) K (capital) is crystal momentum. For such transition, photon provides most of the energy transfer (and a little momentum transfer hk, k in lower case), phonon provides most of the momentum transfer (and a little energy). Similar graphs can be found in most solid state physics textbooks. The picture tells me, either the photon participating in the transition carries crystal momentum, which value is equal to physical moemntum hk, or the crystal momentum itself is a kind of physical momentum. However, one can prove that a phonon does not carry physical momentum (here I quote Kittel's "Introduction to Solid State Physics"): ![](http://i.stack.imgur.com/AohrD.jpg) So, how do we explain the momentum transfer in the electron transition aforementioned?


title: Should you really lean into a punch?

text: There's a conventional wisdom that the best way to minimize the force impact of a punch to the head is to lean into it, rather than away from it. Is it true? If so, why? EDIT: Hard to search for where I got this CW, but heres one, and another. The reason it seems counter-intuitive is that I'd think if you move in the direction that a force is going to collide into you with, the collision would theoretically be softer. You see that when you catch a baseball barehanded; it hurts much more when you move towards the ball, rather than away from the ball, as it hits your hand.


title: Universality of Quantum Operations

text: Does an analog of the Solovay-Kitaev theorem exist for quantum operations, a generalization of quantum gates that also includes all completely positive maps?


title: Calculate time to heat my swimming pool

text: I have a swimming pool of 5300Liters. I have a heating element from a washing machine taking 1min17s to heat a bucket (10L) of water from 22 degrees to 30 degrees celcius. the water in my pool is 19 degrees celcius. how long would it take to warm my pool to 30 degrees? How long would it take to warm it to 25 degrees? is there any more information you need?


title: Degeneracies of the first excited state

text: I have a box with $x,y,z$ all ranging from 0 to $l$. It has $V(x)$=0 inside and =$\infty$ outside. By extending the 1D Schrodinger equation, I have that the allowed energy eigenvalues are $\hbar^2\pi^2\over2ml^2$$(n_1^2+n_2^2+n_3^2)$. What is the degeneracy of the 1st excited energy level? By "1st excited energy level" does that mean 1 of the $n_k$'s, say $n_1$, =2 while $n_2=n_3=1$? Or does it mean all 3 dimensions are in their 1st excited state -- $n_k=2$ $ \forall k\in${$1,2,3$}? Also, how does one find the degeneracy? I am guessing that it is 3? Thanks for any help.


title: Bubble formation

text: I know that surface tension plays a key role in the formation of a bubble. I guess a bubble contains air inside it. Now how is it so that a soap bubble contains air both inside it and outside it? I will be grateful to an answer donor if he explains me the various stages of bubble formation. In brief "How to make a bubble from a glass of water, what are the various processes that take place during the bubble formation"??


title: What is use of inertial frames

text: In class of Newton's laws of motion, it was explained that Newton's second law is valid only in inertial frames. Teacher give us a example by considering a lift which is going downwards with acceleration$=a$ then a man watching the lift from outside will write $mg-t=ma$ (where $m$ is mass of the lift and $t$ is upward force on the lift) while a man inside the lift will write $mg-t=0$ (for him there is no acceleration of lift) both equations will give us different value of $t$ so we have to have a frame specified for work ie inertial frame but nearly after 5 months I have a problem. In the observation of the man inside lift acceleration due to gravity is stated as $g$ but I think that it should have value $g-a$ (if a stone fall on Earth with acceleration $g$ then inside the lift it should fall with acceleration $g-a$) and then by substituting this value for acceleration due to gravity in equation of man inside lift we will get the same value of $t$ as we got in equations of man outside the lift. So if both frames give same value of $t$ then why we prefer one frame after other.


title: How do I approach this conservation of energy problem, symbolically

text: In the absence of nonconservative forces such as friction and air resistance, the total mechanical energy in a closed system is conserved. This is why that when I toss an object directly upwards, the kinetic energy $K = (1/2)mv^2$ is transformed into potential energy as it increases in height with potential energy $U = mgh$ Because of the conservation due to the energies being transformed, we can express this relationship between the two energies as $K_i + U_i = K_f + U_f$ The question I was asked was to use these equations to find the maximum height $h_{max}$ to which the object will rise, as expressed in terms in $v$ and $g$. I was able to solve this by saying that at this max height, the velocity and therefore the kinetic energy will be at _zero_. So I am able to say that $K_i + 0 = 0 + U_f$ or simply $K_i = U_f$ Mass cancels and we are left with $\frac{v^2}{2g} = h_{max}$ This was easy enough, and it is likely that my misunderstanding is simply a mathematical one, but I am at a loss when asked > At what height $h$ above the ground does the projectile have a speed of > $0.5v$ How do you approach this problem?


title: Is terahertz radiation new?

text: Regarding the future applications of terahertz technology, I thought we already knew the entire electromagnetic spectrum. In fact we do not, terahertz lies between microwaves and infrared radiation. The future applications of this is astonishing. For example, faster communications and next generation scanning and imaging technologies. The TSA will pretty much be out of a job, because T-Ray(terahertz) scanners will automatically detect chemical weapons and bombs. I know this because they leave a trace on the terahertz spectrum. The downside with communications is terahertz is absorbed easily by the Earth's atmosphere, so the communications cannot be too high in the sky. What I am asking is, are there any other frequencies on the electromagnetic spectrum we might discover? I was taught in school, that there are radio waves, microwaves, infrared, visible, ultra violent, x-rays, and gamma rays. Now the books have to be rewritten because of the existence of terahertz or t-rays. Remember it lies right between microwaves and infrared radiation. Another question is what about sound? So far there is infra sound, sound, and ultrasound. Do you think in the future we will discover other types of sound? Sorry for the length of my question, as you can see I know a lot about science and technology.


title: Solving quantum radial equation for infinite potential spherical annulus for $l=0$

text: There is a mass $m$ in a potential such that $$ V(r) = \left\\{ \begin{array}{lr} 0, & a \leq r \leq b\\\ \infty, & \text{everywhere else} \end{array} \right. $$ I am looking to find the solution $u(r)$ to the radial equation $$ -\frac{\hbar^2}{2m} u^{\prime \prime}(r) + \bigg( V(r) + \frac{\hbar^2}{2m} \frac{l(l+1)}{r^2}\bigg) u(r) = E\, u(r)$$ in the case $l=0$. $\textbf{Progress so far}$ Looking at the region with zero potential, and letting $l=0$, I define $$ k \equiv \frac{\sqrt{2mE}}{{\hbar}} $$ so that I have the second order differential equation $$ u^{\prime \prime}(r) = - k^2 u(r)$$ which has the solution $$u(r) = A \sin(kr) + B \cos( kr) $$ The potential is such that the boundary conditions are $$ u(a) = u(b) = 0 $$ $$ A \sin(ka) + B \cos(ka) = A \sin(kb) + B \cos(kb) $$ It seems that, since, for instance, $\sin(kb) = \sin(ka)$, $$kb = ka + 2\pi n$$But this must be wrong because, for instance, $$\sin\bigg (\frac{2 \pi n a}{b - a} \bigg ) \neq 0 .$$ Could someone tell me where I went wrong in finding $k?$


title: How light years is measured.?

text: How light years are measured. Once i remember the distance between earth and moon are measured by light which travels and comeback and by the delay the distance is calculated.. But how light years are calculated


title: Why doesn't current flows in reverse biased diode?

text: Consider this reverse biased diode : ![enter image description here](http://i.stack.imgur.com/z9dZo.png) I read that no or very small current flows in reverse biased diode as depletion layers get widened and huge resistance is offered so no electrons can cross it. But, why the electrons or holes need to cross the depletion layer? In the diagram above, the positive charges (holes) are moving towards left and the current due to electrons is also in left, so won't the circuit be completed?


title: What are bootstraps?

text: I've heard occasional mentions of the term "bootstraps" in connection with the S Matrix. I believe it applies to an old approach that was tried in the 1960s, whereby - well I'm not sure - but it sounds like they tried to compute the S Matrix without the interaction picture/perturbation theory approach that we currently use. I'm aware that the approach was abandoned, but my question is: how was it envisaged to work ? What was the input to the calculation supposed to be and how did the calculation proceed ? I know it's got something to do with analyticity properties in terms of the momenta, but that's all I know.... For example, from the wikipedia article: > Chew and followers believed that it would be possible to use crossing > symmetry and Regge behavior to formulate a consistent S-matrix for > infinitely many particle types. The Regge hypothesis would determine the > spectrum, crossing and analyticity would determine the scattering > amplitude--- the forces, while unitarity would determine the self-consistent > quantum corrections in a way analogous to including loops. For example - I can't really understand how you would hope to compute the scattering amplitude just given crossing symmetry and assuming analyticity


title: Solving the 1-D Schrodinger equation for a free particle: Confused about 2 possible general solutions

text: I am following Griffiths' Introduction to Quantum Mechanics, as well as an online lecture that follows a different book, and both sources give different equations for the general solution of the 1-D Schrodinger equation for a free particle. Griffiths has it as: $$\Psi(x,t)=\frac{1}{\sqrt {2\pi}}\int_{-\infty}^{+\infty}\phi(k) e^{ i\left(kx-\frac{hk^2}{2m}t\right)}dk\tag1$$ The online lecture has it as:$$\Psi(x,t)=\frac{1}{\sqrt {2\pi\hbar}}\int_{-\infty}^{+\infty}a(p) e^{ \frac{i}{\hbar}\left(px-\frac{p^2}{2m}t\right)}dp\tag2$$ The most glaring addition to the second equation is the $\hbar$ under the square root sign. Given this, I can't see how they're both equivalent. The link to the online lecture is http://www.youtube.com/watch?feature=player_detailpage&v=xm-LMpkqSUQ#t=1273s


title: Change of QM Momentum operator under coordinate transformation

text: Can any one please let me know what is the general procedure to construct the momentum operator under some coordinate transformation? For example, I understand that if $${\bf{r}}\rightarrow{\bf{r'}}=a{\bf{r}}$$ then the momentum operator will change as: $$\frac{\partial}{\partial{\bf{r}}}\rightarrow\frac{1}{a}\frac{\partial}{\partial{\bf{r}}}.$$ However, what if the factor $a$ itself if a function of ${\bf{r}}$. My understanding is that in such a case, the differential $d{\bf{r'}}$ may be written as $$d{\bf{r'}}=a\ d{\bf{r}}+{\bf{r}}\ da({\bf{r}}).$$ Then, is there any procedure to calculate how would the momentum operator transform?