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<p>The title says it all: is there a table that sums up the parameters, the assumptions/symmetries, and the (most important) predictions of the <a href="http://en.wikipedia.org/wiki/Standard_Model" rel="nofollow">standard model</a>? </p>
| 7,876 |
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<p>Consider air from outside at $T_0$ effusing into an evacuated and thermally isolated chamber.</p>
<p>By thermodynamic potentials, the temperature inside the tank is given by:</p>
<p>$$T_1 = \gamma T_0 = \frac{5}{3} T_0$$</p>
<p>However, by using Maxwell-Boltzmann statistics, the distribution of particles effusing is $\propto v^3 exp(-\alpha v^2)$. Calculating the average KE, we find that</p>
<p>$$T_1 = \frac{4}{3}T_0$$</p>
<p>Why is there a difference between the two?</p>
| 7,877 |
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<p>A particle is projected with velocity $u$ from the bottom of an inclined plane whose inclination with the horizontal is $\beta$. If afterwards the projectile strikes the inclined plane perpendicular to the surface, find the height of the point struck (distance from the ground). The angle made by the velocity with the incline is $\alpha$.</p>
<p>I got an answer while solving this question that differs from the one in my textbook. I got the correct value for the time period though... This is what I did:</p>
<p><img src="http://i.stack.imgur.com/1F2MI.jpg" alt="enter image description here"></p>
<p>Could someone please try this question? Please help me find the height and also tell me the expression you get for the time taken.</p>
<p>My textbook shows</p>
<p><img src="http://i.stack.imgur.com/7ZqKe.jpg" alt="enter image description here"></p>
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<p>When doing integration over several variables with a constraint on the variables, one may (at least in some physics books) insert a $\delta\text{-function}$ term in the integral to account for this constraint.</p>
<p>For example, when calculating
$$\displaystyle\int f(x,y,z)\mathrm{d}x\mathrm{d}y\mathrm{d}z$$
subject to the constraint $$g(x,y,z)=0,$$ one may calculate instead
$$\displaystyle\int f(x,y,z)\delta\left(g(x,y,z)\right)\mathrm{d}x\mathrm{d}y\mathrm{d}z.$$</p>
<p>The ambiguity here is that, instead of $\delta\left(g(x,y,z)\right)$, one may as well use $\delta\left(g^2(x,y,z)\right)$, $\delta\left(k g(x,y,z)\right)$ where $k$ is a constant, or anything like these to account for the constraint $g(x,y,z)=0$.</p>
<p>But this leads to problems, since the result of the integration will surely be changed by using different arguments in the $\delta\text{-function}$. And we all know that the $\delta\text{-function}$ is not dimensionless.</p>
<p>My impression is that many physical books use the $\delta\text{-function}$ in a way similar to the above example. The most recent example I came across is in <a href="http://books.google.com/books?id=DTHxPDfV0fQC&lpg=PP1&pg=PA3#v=onepage&q&f=false" rel="nofollow">"Physical Kinetics"</a> by Pitaevskii and Lifshitz, the last volume of the Landau series.</p>
<p>In their footnote to Equation (1.1) on page 3, there is a term $\delta(M\cos\theta)$ to account for the fact that the angular momentum $\mathbf{M}$ is perpendicular to the molecular axis. But then, why not simply $\delta(\cos\theta)$ instead of $\delta(M\cos\theta)$?</p>
<p>One may say that when using $\delta(\cos\theta)$, the dimension of result is incorrect. Though such an argument may be useful in other contexts, here for this specific example the problem is that <strong>it is not clear what dimension should the result have</strong> (the very reason for me to have this question is because I don't quite understand their Equation (1.1), but I am afraid that not many people read this book).</p>
<p>To be clear: I am not saying the calculations in this or other books using the $\delta\text{-function}$ in a way similar to what I show above are wrong. I am just puzzled by the ambiguity when invoking the $\delta\text{-function}$. What kind of "guideline" one should follow when <strong>translating</strong> a physical constraint into a $\delta$-function? Note that I am <strong>not</strong> asking about the properties (or rules of transformation) of the $\delta\text{-function}$.</p>
<p><strong>Update</strong>: Since this is a stackexchange for physics, let me first forget about the $\delta\text{-function}$ math and ask, how would you understand and derive equation (1.1) in the book "Physical Kinetics" (please follow <a href="http://books.google.com/books?id=DTHxPDfV0fQC&lpg=PP1&pg=PA3#v=onepage&q&f=false" rel="nofollow">this link</a>, which should be viewable by everyone)?</p>
| 7,879 |
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<p>I know that $p=mv$ and (0.1kg)(15m/s)=1.5 kg m/s and the momentum at the vertex is 0, but what is the momentum halfway up?</p>
| 7,880 |
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<p>I recently discovered this website <a href="http://www.quantumphil.org/" rel="nofollow">http://www.quantumphil.org/</a> and wondering whether <a href="http://www.google.com/search?as_q=quantum+philosophy" rel="nofollow">Quantum Philosophy</a> is an actual field, or just an aspect of QM?</p>
<p>Apologies if this is in the wrong place.</p>
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<p>I have 1 litre of air in a sealed container at atmosphere pressure (approx 15 psi absolute).
I want to reduce pressure in the container by 1 psi ( 14 psi absolute or -1 psig). </p>
<ul>
<li>If I understand correctly that 1 psi reduction is 6.66 % reduction in pressure compared to standard atmospheric pressure, </li>
<li>Applying the same to volume would it be correct to say that
if I increase the volume of the container by 6.66 % the pressure would drop by 1psi for air OR</li>
<li>If i remove 6.66% volume of air in the container I would still be able to achieve the 1psi drop in pressure?</li>
</ul>
<p>Do some other factors come in to play for pressure / volume calculation...? </p>
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<p>Force is explained as a push or pull, feels quite intuitive at first. </p>
<p>My question is if an object accelerates over distance of $100$ meter hits a car, and another object of same mass at same acceleration over a distance of $1000$ meters hits the same car, then the force of these two objects is the same, since mass and acceleration of the two objects are the same. Yet, the "push" felt would be very different, because surely the second object will "push" the car much "harder" and "farther" than the first. </p>
<p>Applying another physic equation, work, forced over distance, may explain this, but I am left not truly understanding what/where is force in everyday situations - how can force be understood as a "push" if the push felt is different from force of the same magnitude? </p>
<p>My second question is the basic math for Newton unit. My understanding of multiplication is a way of counting: unit per group multiplied by number of groups yielding total number of units. How/what's the best way to understand the math of physics composite units, such as force: the product of $\text{kg}$ and $\text{m/s$^2$}$ yielding a composite unit, $\text{N}$? Within the context/definition of multiplication, What am I actually counting with two different mixed units? </p>
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<p>Can we destroy a <a href="http://en.wikipedia.org/wiki/Black_hole" rel="nofollow">black hole</a>? I know it is pretty powerful and hardcore stuff but there should be a way even theoretically as we know know nothing is perfect in this universe as it exist due to imperfection. Then how could a black hole be so perfect, so near to god power? </p>
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<p>Where $\triangledown$ is the covariant derivative and we are to assume that the connection is torsion free (that is, we can exchange the lower indices of the connection coefficients), how can I prove the following identity?</p>
<p>$$
X^b\partial_bY^a-Y^b\partial_bX^a=X^b(\triangledown_{{\partial}_b}\overrightarrow{Y})^a-Y^b(\triangledown_{{\partial}_b}\overrightarrow{X})^a
$$</p>
<p>I am struggling quite a bit with this. If anyone has any input on how to show this, please let me know.</p>
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<p>There is a strong physics case for building and running a muon collider (primarily as a Higgs factory).</p>
<p>Currently, what is the principal technological hold-up in making the muon collider possible?</p>
| 7,886 |
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<p><a href="http://en.wikipedia.org/wiki/Ballistic_conduction" rel="nofollow">Ballistic Conduction</a> is the phenomenon of an ideal conduction environment for quantum particles - for electrons the Ballistic Conduction is not infinity, but is proportional to the difference between Fermi surfaces of the leads on both sides.<br>
One of the reasons, it seems to me, is that electron is a Fermion and cannot occupy the "transport state" that spans the length of the leads - if there is another electron occupying it. <a href="http://rads.stackoverflow.com/amzn/click/0521599431" rel="nofollow">Supriyo Dutta</a> also argues for this in terms of chemical potential.<br>
It seems to me then that for bosons like phonons or photons, if there exists even one state spanning the leads, there should be an effective short circuit and the transmission would be the theoretical maximum. Is that true?</p>
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<p><img src="http://i.imgur.com/hXoK3PP.png" alt="question">
<img src="http://i.imgur.com/9o7hB3h.jpg?1" alt="answers"> </p>
<p>I have come up with the answers to parts a and b as shown in the image above. However in part c, I can't seem to find the answer, even though it only involves fitting numbers into Bernoulli's equation.
$P+\frac{\rho u^2}{2}+\rho g z=const$
Is this because I am using the wrong relative velocity?</p>
| 7,888 |
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<p>I know that universe is expanding equally between every pair of points but it was a single point in it's very past... so I was wondering if we could locate this center point of universe. Now I know that we can't and read (on this site) that whatever direction would we go - it would take us closer and closer to big bang... it makes me wonder - if we will trying to get there, there will be our new starting point so going back should take us even closer to this big bang but well, we should have the opportunity to go back to our 1st starting point and it was "farther" before, not closer at all.</p>
<p>Also, as the universe is still expanding and the stars will eventually run out of fuel, would it be possible to try to escape from the situation by flying ahead the big bang and reaching parts of universe that have their state similar to past of "our part of the universe"? Does it mean that the universe is creating itself all the time and dying all the time too?</p>
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<p>Is Wick rotation invariant under proper conformal transformations? Why or why not?</p>
<p>Does Wick rotation apply to conformal field theories? $(1-i\epsilon )T$ is not invariant under proper conformal transformations.</p>
| 7,890 |
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<p>In this (schematic) equation to calculate the scattering amplitude A by integrating over all possible world sheets and lifetimes of the bound states</p>
<p>$$ A = \int\limits_{\rm{life time}} d\tau \int\limits_{\rm{surfaces}} \exp^{-iS} \Delta X^{\mu}
(\sigma,\tau)$$</p>
<p>the information about the incoming and outgoing particles is still missing. It has to be inserted by hand by including additional multiplicative factors (vertex operators)</p>
<p>$$ \prod\limits_j e^{ik_{j_\mu} X^{\mu}(z_j)}$$</p>
<p>into the integrand as Lenny Susskind explains at about 1:18 min <a href="http://www.youtube.com/watch?v=s43SMaZHa50" rel="nofollow">here</a>. But he does not derive the fact that the information about the external particles can (or has to) be included by these additional multiplicative factors like this, he just writes it down.</p>
<p>Of course I see that these factors represent a particle with wave vector $k$, and $z$ is the location of injection (for example on the unit circle when conformally transforming the problem to the unit disk) over which has finally to be integrated too.</p>
<p>But I'd like to see a more detailed derivation of these vertex operators (do there exit other ones too that contain additional information about other conserved quantities apart from the energy and the momentum?) and how they go into the calculation of scattering amplitudes, than such "heuristic" arguments.</p>
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<p>I'm a private pilot, and I have some questions to those who have knowledge of the aeroelastic effects and flutter phenomenon. I would like to talk a little about aerodynamic flutter onset speed and flight control malfunction. It is known that freeplay, worn-out control rods or slop in flight control cables might induce flutter.</p>
<p>What I'm interested in is how critical flutter speed is affected by those problems. I'm wondering especially about a cable control failure where the surface would be disconnected and freefloating. There are small light sport aircrafts and even some FAR 23 standard certified aircrafts which don't have mass-balanced surfaces, especially ailerons which I guess would be more prone to flutter. How critical flutter speed lowers in a situation like that (and how prone to violently flutter are these ailerons in an emergency disconnected sitation)? Is there a linear drop in flutter speed? Can it reach even lower speeds in the normal operating envelope e.g. below usual cruise speed? Normally, assuming no malfunction, flutter speed is at least 10% above Vne (never exceed speed) or Vdf (max dive test speed used during certification flight test). How do you think these things might change?</p>
<p>The same about the others control surfaces e.g. a broken trim tab linkage.</p>
<p>I really appreciate your help. Thank you very much!</p>
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<p>I'm aware that we can describe the time evolution of states/operators (choose your favourite picture) of non interacting quantum fields and that perturbation theory is very effective in computing S matrix elements between free states in the remote past and free states in the remote future. Clearly the non-perturbative description of what's going on at finite times, where the interaction is active is intractible, but my question is - are there simplified toy models (scalar electrodynamics ? reduced numbers of dimensions ?) where we <em>can</em> describe what's happening non perturbatively.</p>
<p>Even if nothing like electrodynamics has been treatable like this, any results on the other "textbook" QFTs (like $\phi^4$) would be interesting.</p>
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<p>Would a disconnected surface, especially <a href="http://en.wikipedia.org/wiki/Aileron" rel="nofollow">aileron</a>, deflect upwards as you slow down due to increased alpha? I figure out it is more likely to deflect upwards as you increase your airspeed, thus having more airflow pressure below it and pushing it upwards.</p>
| 7,894 |
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] |
<p>Do two resistors in parallel dissipate more heat per unit time for an applied voltage when compared to two resistors in series?</p>
| 7,895 |
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<blockquote>
<p>We receive sunlight on earth surface. What type of light beams are these?</p>
<p>Random/Parallel/Converging/Diverging</p>
</blockquote>
<p>I think it should be Diverging as Sun is radiating these beams away. But in one book, answer is given as Random, in another it's Parallel.</p>
| 7,896 |
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<p>I'm learning about accelerating reference frames (to eventually get grasp of general relativity too).</p>
<p>I've just read about the <a href="http://en.wikipedia.org/wiki/Rindler_coordinates#A_.22paradoxical.22_property" rel="nofollow">Rindler coordinates</a> and this one caught my eye</p>
<blockquote>
<p>Note that Rindler observers with smaller constant x coordinate are accelerating harder to keep up! This may seem surprising because in Newtonian physics, observers who maintain constant relative distance must share the same acceleration. But in relativistic physics, we see that the trailing endpoint of a rod which is accelerated by some external force (parallel to its symmetry axis) must accelerate a bit harder than the leading endpoint, or else it must ultimately break. This is a manifestation of Lorentz contraction. As the rod accelerates its velocity increases and its length decreases. Since it is getting shorter, the back end must accelerate harder than the front.</p>
</blockquote>
<p>So does this mean, that in Einstein's elevator (that's used to introduce general relativity, and is supposed to accelerate uniformly) the acceleration felt by the observer on the top is smaller than the acceleration felt by the observer at bottom?</p>
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] |
<p>Can anybody explain to me the <a href="http://en.wikipedia.org/wiki/Coand%C4%83_effect" rel="nofollow">Coandă effect</a> well? I am finding that many definitions and explanations conflict.</p>
<p>I am particularly confused about the following points regarding the Coandă effect:</p>
<ul>
<li>whether the Coandă effect is the same as normal boundary layer attachment to a surface</li>
<li>whether the Coandă effect is only defined for convex surfaces over which a jet (of the same state as the surrounding fluid) is tangentially blown</li>
<li>what causes the Coandă effect: the jet entraining the ambient fluid or a balance of pressure and centrifugal forces, or something else?</li>
</ul>
| 7,898 |
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<p>I have been planning on building a solar grill for quite a time now, and the one problem, which is also the biggest - I need to shape a mirror into a paraboloid, so it bundles solar energy efficiently.</p>
<p>My designs for the whole thing have the focal area of the paraboloid - i.e. the grillage - placed in close proximity of the mirror, depending on the design between 1 and 1.20 meters.</p>
<p>I am afraid that the heat radiating from the focal area would heat up the reflective surface too much, especially if I use an easy-to-shape material like acrylic mirror, which would without a doubt deform after a couple of minutes.</p>
<p>So long story short, I want to know if there is a material that</p>
<ol>
<li>Has high reflective properties</li>
<li>Does not deform at temperatures of 300°C or even higher (400°C to be on the safe side, if the BBQ is going on for hours and to calculate in unexpected risks)</li>
<li>Is reasonably easy to shape (it should be possible without having to handle gigantic machinery or overly dangerous tools)</li>
<li>Can be paid for by a poor engineering-student, so the amount I need should be below 200$. If possible. (a griller is pointless if I have to eat rice for the rest of the year)</li>
</ol>
<p>Ps.: English is not my first language, so there likely are a lot of mistakes in there. If something is unclear, please comment! I apologize for those mistakes beforehand. </p>
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<p>Before the question: I am working on numerical calculation of three dimension parabolic equation that based on <a href="http://en.wikipedia.org/wiki/Thermal_conduction#Fourier.27s_law" rel="nofollow">Fourier's Law</a> of which I am a little confused.</p>
<p>Here comes the law in modern mathematics language.</p>
<blockquote>
<p>"The local heat flux is proportional to temperature gradient"
$$
\vec{q}=-k\nabla T,
$$
where $k$ is the material's conductivity.</p>
</blockquote>
<p>How extremely concise it is, but how to understand the Law? I read the book written by Fourier in 1822 but I know neither of the law in modern mathematics language nor in Fourier's language. I found that every statement or formula related to proofing the Law are not done with rigour enough.
Here is some statement from a <a href="http://catatanabimanyu.files.wordpress.com/2011/09/heat-transfer-_yunus-a-cengel_-2nd-edition.pdf" rel="nofollow">book</a> by YUNUSA.CENGEL on its page 65 chapter 2.</p>
<blockquote>
<p>To obtain a general relation for Fourier's law of heat conduction, consider a medium in which the temperature distribution is three-dimensional. The figure below shows an isothermal surface in that medium. <strong><em>The heat flux vector at a point $P$ on this surface must be perpendicular to the surface, and it must point in the direction of decreasing temperature.</em></strong> If $n$ is the normal of the isothermal surface at point $P$, the rate of heat conduction at that point can be expressed by Fourier's law as
$$
\dot{Q_n} = -kA\cfrac{\partial T}{\partial n}
$$</p>
</blockquote>
<p><img src="http://i.stack.imgur.com/Aa8d5.jpg" alt="enter image description here"></p>
<p>My questions toward the issue I have mentioned are</p>
<ul>
<li>How could heat flux be a vector?</li>
<li>What's the meaning of the direction of the heat flux?</li>
<li>Why heat flux at a point is normal to the isothermal surface?</li>
<li>What's the definition of heat flux vector, not heat flux which is defined as the quantity per second per area?</li>
</ul>
<p>You might say it is true just because of the Second law of thermodynamics.</p>
<blockquote>
<p>The heat always flows spontaneously from regions of higher temperature to regions of <strong>lower temperature</strong>, and never the reverse, unless external work is performed on the system.</p>
</blockquote>
<p>It is <strong>lower</strong> but not <strong>fastest decrease</strong>, isn't?</p>
<p>If the direction is not through the line in the tangent plane of the isothermal surface it would transfer to a colder palce, isn't? So why choice the normal line to be the heat flux direction since there is infinity line to the colder place. Maybe it is that project works when considering the other line! However, it is human beings not the nature define the direction of heat flux for convenient. Am I right?</p>
<p>It may be related with Fick's Law. I am not sure about the proof of three dimension situation.</p>
| 7,900 |
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<p>The <a href="http://en.wikipedia.org/wiki/Weibull_distribution" rel="nofollow">Weibull distribution probability density</a> is given by:
$$f(w,k,\lambda)=\begin{cases}\frac{k}{\lambda}\left(\frac{w}{\lambda}\right)^{k-1}e^{-\left(w/\lambda\right)^k} &w\geq0 \\ 0 & w<0\end{cases}$$</p>
<p>How can I modify the Weibull distribution to apply in a location where the wind is dead calm ($w = 0$) a fraction $\nu$ of the time.</p>
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<p>In quantum field theory, the propagator $D(x-y)$ doesn't vanish for space-like separation. In Zee's book, he claims that this means a particle can leak out of the light-cone. Feynman also gives this interpretation.</p>
<p>What is wrong, then, with this thought experiment:</p>
<p>Bob and Alice synchronize their watches and space-like separate themselves by some distance. Bob tells Alice that at exactly 3 o'clock, he intends to test his new particle oven, which claims to make a gazillion particles. Since Bob's oven makes so many particles, there is a pretty good chance that Alice will detect a particle right at 3 o'clock. If she detects such a particle, Bob will have transmitted a piece of information (whether or not the oven works) instantaneously - way faster than the speed of light.</p>
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<p><img src="http://i.stack.imgur.com/qBLOQ.png" alt="enter image description here"></p>
<p>figure from <a href="http://webusers.physics.illinois.edu/~m-stone5/mma/notes/amaster.pdf" rel="nofollow">http://webusers.physics.illinois.edu/~m-stone5/mma/notes/amaster.pdf</a></p>
<p>The string has fixed ends, a mass per unit length of $\rho$, and is under tension $T$. This <a href="http://webusers.physics.illinois.edu/~m-stone5/mma/notes/amaster.pdf" rel="nofollow">source</a> claims that "the rate that a segment of string is doing work on its neighbour to the right" is </p>
<p>$$
-T\dfrac{\mbox{d} y}{\mbox{d} x}\dfrac{\mbox{d}y}{\mbox{d} t}
$$</p>
<p>without any justification. I have no idea how this is derived. I would like to see a simple (if possible) derivation. I wonder in what way can the a segment of the string $dx$ can do work to the neighboring segment on the right since in the context of this problem there is no movement in the x-axis (right?).</p>
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<p>This is a repost of one of my formerly-stated questions, now with correct tags and format:</p>
<p>In reference to this problem:</p>
<blockquote>
<p>An electron is placed into a dielectric container with capacitors at each end. The side of the container is of length l. A small tunnel is hollowed out in the middle, which is where the electron is contained. The whole container has charge Q and the capacitors supply a potential difference $\phi$ across the material. Find the total potential energy (not just electric potential) of the electron if we measure x from the center of the tunnel.</p>
</blockquote>
<p>In lecture, the Professor said that the equation for the potential is</p>
<p>$$\frac{1}{2}Kx^2-\frac{|e|\phi}{l}x$$</p>
<p>where $|e|$ is the magnitude of the electron's charge. However, he neglected to state the derivation as well as what exactly K was in the equation. Can someone please help me understand? </p>
| 7,904 |
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<p>I'm looking for a nice paper that explains the difference between three particle physics models for spin-independent dark matter interaction with nuclei: elastic, inelastic and isospin violating scattering. I've found a nice paper that is giving a nice summation of the current results in direct dark matter searches ( <a href="http://arxiv.org/abs/1210.4011" rel="nofollow">http://arxiv.org/abs/1210.4011</a> ) and want to read up some more.</p>
<p>Cheers,
Adnan</p>
| 7,905 |
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<p>I am having a bit of a crisis in understanding of the physical meanings of total derivatives.</p>
<p>When a quantity $\rho$ (be it a <em>vector</em> or a <em>scalar</em>) is said to be <strong><em>conserved</em></strong>, then (mathematically) $$\frac{d\rho}{dt} = 0$$
(right??)</p>
<p>Now, if I just integrate both sides with respect to time I get $\rho$ = constant.</p>
<p>But the total derivative can be written as (via the chain rule) $$ \frac{d\rho}{dt} = \frac{\partial \rho}{\partial t} + \mathbf{u}\cdot\nabla\rho$$
where I am not even sure what $\mathbf{u}$ is (the speed of a moving reference frame?).</p>
<p>Given these equations, if the former has $\rho$ = constant as a solution, then what is the point of partial derivative and gradients? They are all going to be zero anyway?</p>
<p>And what exactly is the <strong>physical meaning</strong> of the total derivative?
In <em>fluid dynamics</em> and <em>plasma physics</em> I have been told that it describes how a quantity changes when observed from the frame "moving with the fluid"...</p>
| 7,906 |
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<p>Does sound gets faster when air bubble is suspend in water?</p>
<p>c = sqrt(K/P)</p>
<p>c = speed</p>
<p>K = bulk module</p>
<p>P = density</p>
<p>When air bubbles is homogenized into water the density is lower, so should sound gets faster? Thank you.</p>
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] |
<p>Most of the time, temperature evolution for incompressible flow is computed assuming the <a href="http://en.wikipedia.org/wiki/Eckert_number" rel="nofollow">Eckert number</a> is negligible, that is the viscous dissipation and work of pressure variation is negligible compared to convection and diffusion.</p>
<p>However, if a flow is assumed isothermal, work of pressure fluctuation and viscous dissipation are the only terms remaining in the energy conservation equation. In such a case, the isothermal assumption leads to an equilibrium between both and the pressure is ruled by a transport equation where viscous dissipation is a source term. Most of the incompressible numerical solver compute the pressure as the constraint to ensure $\nabla\cdot\mathbf{u}=0$.</p>
<p>How compatible is pressure as a incompressibility constraint with pressure in equilibrium with viscous dissipation?</p>
| 7,908 |
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<p>Suppose that in the intergalactic space far from any significant gravitational attractors there is a relatively small concentration of He-4 atoms. Due to gravitational attraction fermions in this case would form a sphere (gas planet), but as bosons He-4 atoms aren't affected by exclusion prinicple, so what's holding them back from collapsing into singularity?</p>
| 7,909 |
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<p>I know this isn't the right place for asking this question, but in other places the answers are so awfull.. I'm studying eletricity, so, I start seeing things like "charges", "electrons has negative charges",etc. But I didn't quite understand what charge is. I search a little bit on the internet and found it related to electromagnetic fields, then I thought "negative and positive may be associeted with the behaviour of the particle in the field, great!", but the articles about e.m. fields already presuppose "negative" and "positive" charges. In other places, I see answers relating charges to the amount of electrons/protons in an atom, but if that's right, the "negative" electron is an atom without any protons? What about the neutron? So, my questions are (1) What are charges; and (2) How a particle can "be" electrically charged. What does that really mean?
Thanks for your time.</p>
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<p>In representing an electric circuit, we would draw the sense of the current from the positive to the negative pole and the electrons from the negative to the positive .
But as I know electrons' motion create current.
So how is this possible?</p>
| 258 |
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<ol>
<li><p>What is the <a href="http://www.google.com/#q=topological+mass+generation" rel="nofollow">topological mass generation</a> mechanism? </p></li>
<li><p>And what is its relation with the Higgs mechanism? </p></li>
<li><p>Can we say that after the discovery of Higgs boson, the topological mass generation mechanism is ruled out?</p></li>
</ol>
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<p>I'm currently reading Brian Greene's <strong>The Elegant Universe</strong> and have also picked up his other book - <strong>The Fabric of the Cosmos</strong>. In <strong>The elegant universe</strong> Brian makes up an interesting point that empty space is not really empty. This is according to string theory. Now I may be completely wrong, but lets consider the <strong>assumption</strong> that empty space is not really empty.</p>
<p>Now there are several analogies to explain the composition of an atom. One favorite analogy says that the atomic nucleus can be considered to be the size of a football placed in the center of a football field, with the electrons moving just outside this field. <em>That's a lot of space!</em> <strong>Assuming</strong> that empty space is not really empty, what does this atomic space consist of?</p>
| 259 |
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] |
<p><strong>Premises:</strong></p>
<blockquote>
<ol>
<li>The radioactivity is either hastened or slowed inside a fast moving aircraft.</li>
<li>Speed of fastest aircraft: <em>3,529.6 km/h</em>.</li>
<li>The earth's revolution is: <em>107278.87 km/h</em>.</li>
<li>The earth's revolution is significantly higher than the speed of moving aircraft.</li>
</ol>
</blockquote>
<p><strong>Questions:</strong></p>
<blockquote>
<ol>
<li>Are all the premises correct and if not, which part is not correct?</li>
<li>Does this mean that the earth's revolution affects modern dating methods?</li>
</ol>
</blockquote>
<p><strong>Sources:</strong></p>
<blockquote>
<p><a href="http://en.wikipedia.org/wiki/Creation_science#Radiometric_datin" rel="nofollow">Radiometric Dating, Creationism</a>,
<a href="http://en.wikipedia.org/wiki/Flight_airspeed_record#Timeline" rel="nofollow">Aircraft Speed</a>,
<a href="http://geography.about.com/od/learnabouttheearth/a/earthspeed.htm" rel="nofollow">Revolution Speed</a></p>
</blockquote>
| 7,911 |
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<p>What would happen if the earth would stop spinning? How much heavier would we be? I mean absolutely stop spinning. How much does the centrifugal force affect us?</p>
<p>If you give technical answers (please do), please explain them in laymen's terms, thanks.</p>
<p><strong>Edit:</strong> I am asking what would be different if the earth were not spinning. Nevermind the part about it stopping.</p>
| 45 |
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<p>What is the total momentum of the whole Universe in reference to the point in space where the Big Bang took place?</p>
<p>According to my reasoning (and a bit elementary knowledge) it should be exactly equal to 0 since the 'explosion' and scattering of the matter throught the space would not change the total momentum in any way. </p>
| 7,912 |
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<p>I am investigating the relationship between the fall time of a paper tray and its projected area.</p>
<p>In this investigation I have been using the following variables:</p>
<ul>
<li>Controlled Variable: The air density, the height from which we drop the tray, the shape of the tray, its mass and the material it is made from (paper).</li>
<li>Independent Variable: The cross-sectional area of the paper tray.</li>
<li>Dependent Variable: The time it takes for the tray to fall on the ground</li>
</ul>
<hr>
<p>I found the relationship for when the paper tray is traveling at terminal velocity. This relationship I devised by using the formula for the speed of a object traveling at terminal velocity:</p>
<p><img src="http://i.stack.imgur.com/YvjyR.png" alt="Terminal Velocity"></p>
<p>Terminal velocity can be considered uniform motion (acceleration = 0) and so the above formula can be rewritten in the following way:</p>
<p><img src="http://i.stack.imgur.com/47JVL.png" alt="Time Squared vs Projected Area at Terminal Velocity"></p>
<hr>
<p>But all this excludes the acceleration period before a object reaches terminal velocity. I suppose that because the paper tray is only 0.016 kg and the paper tray has a fairly high drag coefficient 1.31 that the acceleration period can be ignored since the tray reaches terminal velocity quite soon, but then again the trays are being dropped from only 2 meters.</p>
<hr>
<p>Can I ignore the acceleration period?
Is there a better equation for the relationship?
Can you suggest further reading?</p>
| 7,913 |
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<p>I am struggling with calculating the exclusive semileptonic $B_c^+\rightarrow J/\psi l^+\nu_l$ decay. I learnt that the amplitude is given by a product of the leptonic current $L^{\mu}$ and the hadronic current $H^{\mu}$
$$
\mathcal{M}(B_c\rightarrow J/\psi l^+\nu_l)=\frac{G_F}{\sqrt{2}}V_{cb}L^{\mu}H_{\mu}
$$
where $V_{cb}$ is the CKM parameter, $L^{\mu}$ and $H^{\mu}$ are expressed as
$$
L^{\mu}=\bar{u}_l\gamma^{\mu}(1-\gamma^5)v_{\nu},\quad
H^{\mu}=\langle J/\psi|J^{\mu}(0)|B_c\rangle
$$
where $J^{\mu}$ is the $V$-$A$ weak current. However, I did not know how this result can be derived. Could anyone provide some help?</p>
<p>There is a second problem. On the tree-level, we have the following Feynman diagram
<img src="http://i.stack.imgur.com/l84vp.png" alt="enter image description here"></p>
<p>If we calculate $\bar{b}\rightarrow\bar{c}l^+\nu_l$ as a three-body decay in the electroweak theory (not the four-fermion approximation adopted above), how does it relate to $B_c^+\rightarrow J/\psi l^+\nu_l$?</p>
| 7,914 |
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] |
<ol>
<li>How much mass would a black hole need to create a Schwarzschild radius that would trap a photon, whereby the photon would (to an outside observer) be continually curved 0.004km/s at the horizon?</li>
</ol>
<p>(assume a non-spinning black hole, then contrast that with a photon traveling in either direction along the equator of a spinning black hole)</p>
<p>Eg.
[<img src="https://drive.google.com/uc?export=download&id=0B7P471Xpxoc1N2FoN1FtRmttV2c" alt="Example">]</p>
| 7,915 |
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<p>What is meant by <a href="http://en.wikipedia.org/wiki/Incompressible_flow" rel="nofollow">incompressible flow</a>?</p>
<ol>
<li><p>The density of the fluid is a constant, $\rho = constant$</p></li>
<li><p>The density of a fluid has a spatial dependence but remains constant in time, $\rho = \rho(\mathbf{r})$</p></li>
</ol>
<p>In both cases $\rho$ should satisfy $\frac{d\rho}{dt} = 0$ right?</p>
<p>(In case 1 is true, how do you call a flow with the properties of 2? How would it be mathematically different?)</p>
| 7,916 |
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<p>I know it's probably the most stupid question there is, but why do they fly are the clouds lighter than air? What's keeping those tiny ice structures floating miles about the ground? I've been looking all over the internet and I can't find acceptable answer to this basic physical question. Can you please help me?</p>
<p>This question was marked as answered - but I don't believe it has been explained satisfactory. Please bare with me on this one.</p>
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<p>If, in a QFT of a scalar field $\phi$, a <a href="http://en.wikipedia.org/wiki/Fock_space" rel="nofollow">Fock space</a> $n$-particle position eigenstate $\lvert x_1\cdots x_n\rangle $ is given by
$$
\lvert x_1\cdots x_n\rangle =\hat\phi^\dagger(x_1)\cdots\hat\phi^\dagger(x_n)\lvert 0\rangle \,,
$$
where $\lvert 0\rangle $ is the vacuum state, then we have
$$
\langle x_1\cdots x_n\lvert\phi\rangle =\phi(x_1)\cdots\phi(x_n)\langle 0\lvert\phi\rangle \,,
$$
with $\hat\phi(x)\lvert\phi\rangle =\phi(x)\lvert\phi\rangle $. </p>
<p>Now, the question is what is the value of $\langle 0\lvert\phi\rangle $? </p>
<p>If there's a vacuum configuration state $\lvert\phi_0\rangle = \lvert0\rangle$, then $\lvert 0\rangle $ is orthogonal to any other configuration in the basis, and we would have
$$
\langle x_1\cdots x_n\lvert\phi\rangle = 0
$$
except for the case $\langle 0\lvert 0\rangle $ and thus a joined basis $\{\lvert x_1\cdots x_n\rangle ,\lvert\phi_0\rangle \}$ for a space of higher dimension than the Fock space. So, I conclude there's no vacuum configuration. But the zero configuration, $\phi(x)=0$, is anyway orthogonal to every Fock space state $\lvert x_1\cdots x_n\rangle $ different from $\lvert 0\rangle $, and thus excludes the presence of any particle. In other words,
$$
\lvert\phi(x)=0\rangle=\lvert0\rangle\langle0\lvert\phi(x)=0\rangle
$$
So, the zero field configuration corresponds to the vacuum state, a contradiction; what am I missing? What's wrong in all this?</p>
| 7,917 |
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<p>I have a simple question: Does gravity slow down a horizontally thrown baseball?</p>
<p>Assuming when a baseball is thrown it has a vertical velocity as well, does it slow does the ball?</p>
<p>Any help is much appreciated.</p>
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<p>I have a question about Noether's theorem in the context of QM, which I'll state in the context of the weak interaction but the basic point could be generalized.</p>
<p>According to Noether's theorem, given an $n$-dimensional Lie group there will be $n$ conserved quantities. $SU(2)$ is 3-dimensional, so that we'd expect 3 such quantities. However, elsewhere the conserved quantities are defined as the 'good quantum numbers', where these are defined as the eigenvalues of the maximal number of commuting generators in the group. In this case, there is just one such generator, and so it seems only one conserved quantity.</p>
<p>Can any one tell me where I'm going wrong? </p>
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<p>The problem was posed as follows. Given a pendulum of length $L$ with a mass $m$ attached to it, which forms an angle $\theta$ from the y-axis to the direction of swinging. </p>
<p>First we had to find the potential energy as a function of the angle, which was trivial enough.</p>
<p>$$U = mgL(1 - cos(\theta))$$</p>
<p>But the next question was to develop the gradient for this potential energy function via the arc length $s = L\cdot\theta$, where $s$ is the arc length.</p>
<p>I calculated the gradient by using the trigonometric definition of $\cos(\theta)$, but I developed the gradient by changing the variable of my function $U(\theta)$ to $U(x,y)$ after which I calculated it further, and found that $F_{tan} = -mg\sin(\theta)$, as desired.</p>
<p>But is there an alternative way to calculate it, by using the gradient, and by utilizing the arc length formula above? The question explicitly asks to develop the gradient via the arc length rather than $x$ and $y$ components.</p>
| 7,920 |
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<p>The <em>kinematic</em> state is defined as the position and orientation in space. The <em>dynamic</em> state is defined as the associated velocities.</p>
<p>What is the correct terminology for the combined kinematic and dynamic state? Can I call it the <em>kinetic</em> state?</p>
| 7,921 |
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<p>Is there any theoretical limitation on the size of the diameter of an Plastic optical fiber? I would like to transmit visible sunlight through it. I see bundles of small cores available but wouldn't it be more efficient to make a very large single core?</p>
| 7,922 |
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<p>Conside a contact interaction given by a delta function on their worldlines. Use a gauge fixed Lagrangian for two point particles in terms of their proper times $t$ and $t^{\prime}$. <strong>Is it possible to find proper equations of motion for this system?</strong>
For e.g.
$$
S=\int dt \; \dot {x}^{2}+\iint\! dt \;dt^{\prime}\; \dot{x}(t)\cdot\dot{y}(t^{\prime})~ \delta^{D}(x(t)−y(t^{\prime}))+\int dt^{\prime} \; \dot{y}^{2}
$$
working in, say, $D$ target space dimensions. This is written as a simplification of the relativistic / curved space system to illustrate the point. I think it will be possible to integrate the delta function out in $D=1$, but not in higher target space dim. The problem I have is in how to do the variation of the delta function term. Physically it is producing an interaction every time the worldlines of the particles intersect and I've tried writing this as a sum over such points - where $x(t_{0})=y(t^{\prime})$ - of $\frac{\delta(t−t_{0})}{\dot{x}(t_{0})}$ but this is valid only in $D = 1$ and I'm still not sure I can get the variation correct.</p>
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<p>I tried many hours to understand the <a href="http://en.wikipedia.org/wiki/Principle_of_least_action" rel="nofollow">principle of least action</a>, and those hours become days... and I still didn't understand that principle/ and how it relates to Newtonian mechanics?</p>
<p>Could someone please explain it to me easily?</p>
<p>Comment: for example if I have in <em>énoncé</em>, light travel between point $a$ and point $b$ in time $s$, we should formulate the equation of motion where the path is the shortest such that it takes $s$ time to travel from point $a$ and point $b$ by that time? Also how does this relate to Newtonian mechanics and how is it better?</p>
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] |
<p>What is the name of this principle? </p>
<blockquote>
<p>If the speed of the centre of mass $\vec{v}_{CM}$ of a solid is constant ($cte$), then the sum of the exterior forces that exerts into this solid $\vec{F}_{ext}$ is zero, and the opposite is true, namely that:</p>
<p>$$\vec{v}_{CM}=cte\Leftrightarrow\sum\vec{F}_{ext}=\vec{0}$$</p>
</blockquote>
| 7,925 |
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<p>The first Navier-Stokes equation (conservation of mass) says:
$\vec \nabla \cdot \vec v=0$</p>
<p>For a stationary flow, the l.h.s of the second equation is (conservation of momentum):
$\rho \frac{D\vec v}{Dt}=\rho (\underbrace{\frac{\partial \vec v}{\partial t}}_{=0} + (\vec v\cdot \vec \nabla) \vec v)=\rho ( \underbrace{(\vec \nabla \cdot \vec v)}_{=0??} \vec v)\stackrel{??}{=}0$</p>
<p>I find that the l.h.s of the conservation of momentum equation is always equal to zero for a stationary field.
I know this isn't true but where am I wrong in this reasoning ?</p>
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] |
<p>Will I get a stronger resonance if I put multiple frequency generators around the same object?</p>
<p><a href="http://i.stack.imgur.com/bdqZW.png" rel="nofollow">example image</a></p>
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<p>The action of a free relativistic particles can be given by
$$S=\frac{1}{2}\int d\tau \left(e^{-1}(\tau)g_{\mu\nu}(X)X^\mu(\tau)X^\nu(\tau)-e(\tau)m^2\right).$$
If we then make an infinitesimal transformation of the parametrization parameter $\tau$ this would be
$$\tau\to\tau^\prime=\tau-\eta(\tau),$$
for an infinitesimal parameter $\eta(\tau)$.</p>
<p>Of course we can describe the system as it pleases us so we know that $$X^{\mu^\prime}(\tau^\prime)=X^\mu(\tau).$$ From this relation we see that $X^{\mu^\prime}(\tau)$ must be
$$X^{\mu^\prime}(\tau)
\approxeq X^{\mu^\prime}(\tau^\prime+\eta(\tau))
\approxeq X^{\mu^\prime}(\tau^\prime)+\eta(\tau)\frac{d X^{\mu^\prime}(\tau^\prime)}{d\tau^\prime}
\approxeq X^{\mu}(\tau)+\eta(\tau)\frac{d X^{\mu}(\tau)}{d\tau}$$
Which all can be summarized as
$$\delta X^{\mu}(\tau)=X^{\mu^\prime}(\tau)-X^{\mu}(\tau)=\eta(\tau)\frac{d X^{\mu}(\tau)}{d\tau}.$$
Now this is all well I hope. But if one does the same argument for $e(\tau)$ one gets the wrong transformation. It is a scalar function so it has to obey
$$e^\prime(\tau^\prime)=e(\tau).$$ Which would give the same transformation.</p>
<p>The right transformation are written in <a href="http://arxiv.org/abs/0908.0333" rel="nofollow" title="David Tong's lectures on string theory">David Tong's lectures on string theory</a> on page 13, eq. 1.10.
The transformation is
$$\delta e(\tau)=\frac{d}{d\tau}\left(\eta(\tau) e(\tau)\right).$$
Could someone show me how this is done and elaborate a little on how one knows how different object transforms?</p>
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<p>I have a question related to the connection between the S-Matrix elements and the path integral formalism. In order to formulate the question, I will just work with a scalar field theory for simplicity. </p>
<p>Let us assume that we are given an action $S[\phi]$. In the path integral formalism, we can now define the generating functional
\begin{equation}
Z[J] \propto \int \mathcal{D}\phi ~ e^{i S[\phi] + \int d^4x~ \phi(x) J(x)}
\end{equation}
and calculate arbitrary vacuum expectation values
\begin{equation}
\left<0| \phi(x_1) \ldots \phi(x_n) | 0 \right>
\end{equation}
using functional derivatives with respect to the source $J$. I also know how to calculate vacuum expectation values in the "canonical quantization formalism" (Wick's theorem etc.). So far so good. </p>
<p>Usually, we are not interested in vevs but rather in S-matrix elements such as $\left<p_1, \ldots, p_n|q_1, \ldots, q_m \right>$ where $p_i$ and $q_j$ are outgoing and ingoing particle momenta. Furthermoe, the transition between $S$-matrix elements and vevs is also clear to me: this is just given by the LSZ reduction formula. So in principle, we are now good to go: we can calculate everything in the path integral formalism and eventually relate this to actual matrix elements using the LSZ formula. </p>
<p>Now come my actual questions:</p>
<ol>
<li><p>It seems that there is a more direct relation between the S-matrix elements and the path integral formalism. In fact, on the Wikischolar article about the <a href="http://www.scholarpedia.org/article/Slavnov-Taylor_identities" rel="nofollow">Slavnov-Taylor identities</a> (written by Dr. Slavnov himself) it is stated that the $S$ matrix can be written as $S = Z[0]$. Where does this come from and how is it to interpret? I am confused because I thought that $S$ was rather a matrix (whose entries, i.e. matrix elements are numbers) and $Z[0]$ is just a number (an evaluated integral). So to me, thsi reads like "matrix = number"... Furthermore, if this equation holds true, how can we obtain the $S$ matrix elements from there? </p></li>
<li><p>Even more confusingly, there seems to be another relation to the $S$-matrix element. I have found this in Weinberg Vol. II, chapter 15.7 around equation (15.7.27). There, we have an action that is of the form $I + \delta I$ (the context is here that $I$ is the gauge fixed action of a non-Abelian gauge theory and $\delta I$ is the change due to a small variation in the gauge-fixing condition, but this does not really matter here). It says then: It is a fundamental physical requirement that matrix elements between physical states should be independent of our choice of the gauge-fixing condition, or in other words, of $\delta I$. The change in any matrix element $\langle\alpha|\beta\rangle$ due to a change $\delta I$ in $I$ is
\begin{equation}
\delta \langle\alpha|\beta\rangle ~\propto ~\langle\alpha|\delta I|\beta\rangle.
\end{equation}
So now, there seems to be even a relation between the action and the $S$-matrix elements. How does this fit into the entire picture? </p></li>
</ol>
<p>My QFT exam is coming up, so thanks a lot for your answers! </p>
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<p>Every now and then, I find myself reading papers/text talking about how this equation is a constraint but that equation is an equation of flipping which satisfies this constraint.
<img src="https://dl.dropboxusercontent.com/u/55765722/0.fawaed/flipping.gif" alt="enter image description here"></p>
<p>Indeed, This equation depends on other equation which is direction of hand .</p>
<p>If we will consider 3 axis (x,y,z) to get the behavior of page , we agree that <code>Z = 0</code> for all particles of the page at default state. </p>
<p>So What about the equation of each point (x,y,z) throughout flipping(browsing) ?</p>
<p>Besides, The position of the particle at a time $t$ is given by,</p>
<p>$$S(t)=\frac{V_0}{a}(1-e^{-at})$$</p>
<p>How to use this equation in this context ? Or is not useful for this context ?</p>
<p>There are : </p>
<ul>
<li>rectilinear motion</li>
<li>uniform rectilinear motion</li>
<li>sinusoidal motion</li>
<li>circular motion</li>
<li>...</li>
</ul>
<p>However, I do not know where I should classify the flipping motion to build the suitable equation which depends on: <code>a</code>(acceleration),<code>V</code>(velocity), <code>t</code>,[<code>X</code>,<code>Y</code> , <code>Z</code>] AND/OR [$$\theta$$] </p>
| 7,930 |
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<p>We've all seen sci-fi movies with asteroid belts that require "great skill" to fly through, but how dense is the asteroid belt really? </p>
<p>How much of the belt could you see from the surface of a given asteroid? </p>
<p>Is it uneven, with dense and sparse patches?</p>
| 7,931 |
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<p>I'm <em>so</em> confused in the use of nuclear masses and atomic masses.
I have two questions.</p>
<p>From the book "Outline of Modern Physics" by Ronald, I understand that the <em>semiempirical mass formula</em> (<a href="http://en.wikipedia.org/wiki/Semi-empirical_mass_formula" rel="nofollow">Weizsäcker's formula</a>) is</p>
<p>$$M=Zm_{p}+(A-Z)m_{n}-b_{1}A+b_{2}A^{2/3}+b_{3}Z^{2}A^{-1/3}+b_{4}(A-2Z)^{2}A^{-1}+b_{5}A^{-3/4} \qquad (1)$$</p>
<p>and this formula is for the dependence of the mass of a nucleus on $A$ (mass number) and $Z$ (atomic number), i.e. is the formula that gives you (approximately) the mass of a <em>nucleus</em>.</p>
<p><strong>1st question</strong> $M$ is the nuclear mass or the atomic mass? (I understand that the difference between these two masses are the masses of the electrons. Obviously it's small, but is nonzero.)</p>
<p>Then, he defines the <em>average binding energy per nucleon</em> like</p>
<p>$$BE=[Zm_{p}+(A-Z)m_{n}-M]c^{2}/A \qquad (2)$$</p>
<p>So far so good.</p>
<p>Then, the book tries to do an example, and calculate the binding energy per nucleon for $_{42}^{98}Mo$ and writes</p>
<p>$$BE=[Zm_{p}+(A-Z)m_{n}-M_{nuc}]c^{2}/A \qquad (3)$$</p>
<p>and says: "where the atomic masses are used for $m_p$ and $M_{nuc}$ (so that the electron masses cancel)." And insert the numbers.</p>
<p>$$BE=[(42(1.007825u)+56(1.008665u)-97.905409u)/98] \times 931.5\frac{MeV}{u} \qquad (4)$$</p>
<p>But, according to <a href="http://en.wikipedia.org/wiki/Isotopes_of_molybdenum" rel="nofollow">Wikipedia</a> the isotopic mass of $_{42}^{98}Mo$ (<em>atomic</em> mass) is 97.9054082 u, and the mass of the <a href="https://en.wikipedia.org/wiki/Proton" rel="nofollow">proton</a> is 1.007276466812 u. So my</p>
<p><strong>2nd question</strong> is: why does the book uses the value 1.007825u for the mass of a proton instead of the value 1.007276466812 u, and why does it uses the atomic mass instead of the nuclear mass in $M_{nuc}$?</p>
<p><strong>Note</strong> The book uses the value 1.007825u for the mass of a proton and the atomic mass instead of the nuclear mass in $M$, in other examples. Why?</p>
| 7,932 |
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<p>In life, when you talk about nuclear energy, there always happens to be a guy who says that famous Einstein's equation. "<em>Yeah, they just convert mass to energy, $E = mc^2$ ya know?</em>" </p>
<p>When I think about that, all I learned about nuclear power resembles dominoes arrangements. You <em>tonk</em> a block and it falls. On its way, it <em>tonks</em> other dominoes and when it falls it releases energy (sound waves).<br>
Quite same in the nuclear physics. You send slow neutron to a core. The core absorbs it, breaks and sends another neutrons and energy (electromagnetic waves). </p>
<p>So in the end, I see no domino blocks disappearing in this game. All we do is, that we <em>tonk</em> domino arrangements that has been built by old stars long time ago. </p>
<p>So why is this equation related to nuclear power? What mass disappears in nuclear power plants?</p>
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<p>Assume the following phase diagram $T-H$ of water</p>
<p><img src="http://i.stack.imgur.com/gU12G.png" alt="enter image description here"></p>
<p>At a nuclear engineering course, I was told that in order to increase the performance of a pressurized water reactor, one has to increase the pressure.</p>
<p>How such a reactor works is seen on the next image</p>
<p><img src="http://i.stack.imgur.com/zrWF9.gif" alt="enter image description here"></p>
<p>In the steam generator the is saturated water. As fusion takes place, very hot water moves towards the steam generator where it transfers its thermal energy to the saturated water. The saturated water becomes hotter and finally it vaporizes. The steam moves the turbine and electric energy is produced.</p>
<p>I cannot understand the following: Why the increment of pressure in the reactor vessel, will transfer more energy in the steam generator making the thermal energy of the steam larger?</p>
| 7,934 |
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<p>It is said in many textbooks that alpha decay involves emitting alpha particles, which are <em>very stable</em>. Indeed, the binding energy (<a href="https://www.oecd-nea.org/dbdata/data/mass-evals1995/mass_rmd_mas95.txt" rel="nofollow" title="Table of atomic masses, including binding energies; OECD Nuclear Energy Agency">~28.3 MeV</a>) is higher than for $Z$-neighboring stable isotopes. But the binding energy is lower than, for example, ${}^9\mathrm{Be}$ (<a href="https://www.oecd-nea.org/dbdata/data/mass-evals1995/mass_rmd_mas95.txt" rel="nofollow" title="Table of atomic masses, including binding energies; OECD Nuclear Energy Agency">~58.2 MeV</a>). My question is why aren't other nuclear compounds ejected from heavy nuclei, e.g. ${}^9\mathrm{Be}$?</p>
<p>The <a href="http://en.wikipedia.org/wiki/Gamow_factor" rel="nofollow">Gamow factor</a>
$$e^{-\frac{4\pi}{\hbar}\frac{Ze^2}{4\pi\epsilon_0}\frac{1}{v_\alpha}}$$
decreases exponentially in $Z$, so it explains intuitively why lower-$Z$ particles would tunnel more often. Specifically, it would explain why we would see ${}^9\mathrm{Be}$ emission ${e^{-2}}\simeq 0.14$ times as often compared to ${}^4\mathrm{He}$ emission. Also, the particles need to form in the nucleus prior emissions; but with a similar binding energy per nucleon (~7.08 MeV for ${}^4\mathrm{He}$ vs. 6.47 MeV for ${}^9\mathrm{Be}$) and higher total binding energy for the ${}^9\mathrm{Be}$ nucleus, I would expect that its formation in the same order of prevalence as the alpha particle (according to Ohanian, between 0.1 and 1 alpha particles are in existence at any moment in time).</p>
<p>Can anyone explain this? A reference to an article/textbook would be preferable.</p>
<p><strong>EDIT</strong>
Same goes for ${}^{16}\mathrm{O}$ which is also a double magic isotope, as 'anna v' pointed out. For it, the Gamow factor is smaller by $e^{-4}\simeq 0.02$, and emission should still be viable.</p>
| 378 |
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<p>Suppose we have the two-loop integral $\int \mathrm{d} ^ 4 k _ {2} \int \mathrm{d} ^ 4 k _ {1} \, f(k _ {1}, k _ {2})$, where $k _ {1}$ and $k _ {2}$ are four-dimensional vectors in Euclidean space. In the first integration with respect to $k _ {1}$, I take $k _ {2}$ to be the z-axis. Then $k _ {2} = k _ {1} \cos \omega$ and the four-dimensional spherical volume element is $\mathrm{d} V = |k _ {1}| ^ {3} \sin ^ 2 \omega \sin \theta \, \mathrm{d} |k _ {1}| \, \mathrm{d} \omega \, \mathrm{d} \theta \, \mathrm{d} \phi$, where $|k _ {1}|$ is the Euclidean norm of $k _ {1}$.</p>
<p>When we perform the second integration $\int \mathrm{d} ^ 4 k _ {2} \, g (k _ {2})$, where $g (k _ {2})$ is the result of the first integration, is it allowed to change coordinate systems and take $k _ {2}$ to be the radius in $S ^ 3$? That is, can we write the spherical volume element for the second integration as $2 \pi ^ 2 |k _ {2}| ^ {3}$, where $|k _ {2}|$ is the Euclidean norm of $k _ {2}$?</p>
| 7,935 |
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<p><strong>Question:</strong></p>
<p>In principle, does a system of gravitational charges exhibit equivalent behavior to a time-reversed system of like electric charges? <em>(At a single instance in time?)</em></p>
<p><strong>Additional Notes:</strong></p>
<p>I am aware that the evolution of this system would not behave the same because orbits cannot manifest in a simple system of like charges due to reasons regarding entropy; Just because it is entropically desirable to evolve from the big-bang state to the current universal state, it is not entropically desirable to evolve from the current universal state to the big-bang state just because we reversed the flow of time. (I intend to reverse the flow of time so that the fields reverse but not the global increase of entropy, this is why I specify <em>At a single instance in time?</em>).</p>
<p>I would ask that SE users to consider that the argument of playing a tape backwards is an ill conceived method for generating an answer since it always admits an unnatural evolution of state of the system (One that would never play out if entropy was increasing, I only reverse time for the sake of reversing the fields). For example if I tape an ink drop falling in water and watch it in reverse, it becomes immediately become apparent to me that it is being played in reverse because the system evolves in a way which violates the second law of thermodynamics. Even though this is true, what I can say with confidence is that every electric field of every particle will reverse in sign and the same can be said about the gravitational field. That is the true purpose of the question. </p>
<p>With that being said, I specify "<em>(At a single instance in time?)</em>" because I am more concerned with this idea on a fundamental level <em>(i.e. The physical properties of the fields)</em>. </p>
| 7,936 |
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<p>Provided that the two pipe lines are of same length, same material and in the same level, is the water pressure in both the layouts same or different?
<br><br><img src="http://i.stack.imgur.com/XVSrc.jpg" alt="enter image description here"></p>
<p>PS: In 1st pipeline the turns are not "upward-downward" turns but "sideward-turns" in same level(height).</p>
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<p>I have read in many places that one point functions, like the one below:</p>
<p>$$\langle \Omega|\phi(x) |\Omega \rangle$$</p>
<p>are equal to zero ( $|\Omega \rangle$ is the vacuum of some interacting theory, $\phi$ is the field operator - scalar, spinorial, etc...)</p>
<p>Peskin's book, for instance, says (page 212) this is USUALLY zero by symmetry in the case of a scalar field ("usually" probably means $\lambda \phi^4$ theory) and by Lorentz invariance for higher spins . How can I see that? </p>
<p>In a more general question: can someone point out a counterexample? A case when this functions are not zero?</p>
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<p>In a potential which needs to be evaluated at the retarded time, is this the time which represents the actual time the "physics" occurred? So $t_{\text{ret}}=t-\frac{r}{c}$, not just because it may be that you are receiving a signal at light speed but because "causality" spreads out at the maximum speed, $c$, is this correct?</p>
<p>The Lienard-Wiechert 4-potential for some point charge ($q$): $A^\mu=\frac{q u^\mu}{4\pi \epsilon_0 u^\nu r^\nu}$ where $r^\nu$ represents the 4-vector for the distance from the observer. In the rest frame of the charge $A^i$ for $i=1,2,3$ is clearly zero but from what has been said about the retarded time we have that $A^0=\frac{q}{4\pi\epsilon_0c(t-r/c)}$.</p>
<p>Obviously I would like to get $A^0=-\frac{q}{4\pi\epsilon_0 r}$, so where is the misunderstanding of retarded time and instantaneous time? Unless we would like the time since the signal was emitted which is $r/c$? Or if $t$ itself is already $t'-r/c$ and we need to return to the instantaneous time $t$, when the signal was emitted.</p>
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<p>In the arrangement shown in the figure below, an object of mass m can
be hung from a string (linear mass density $\mu$ = 2.00 g/m) that passes over
a light (massless) pulley. The string is connected to a vibrator with
constant frequency $f$, and the length of the string between point P and the
pulley is $L = 2.00 m$. When the mass m of the object is either 16.0 kg or
25.0 kg, standing waves are observed in the string.</p>
<p>No standing waves are observed for any other mass between these values.</p>
<hr>
<p>$n$ = number of nodes from 25kg</p>
<p>$n + 1$ = number of nodes from 16kg</p>
<p>I do not understand why they have only integer 1 difference between the numbers of nodes of 25 and 16kg...</p>
<p>Anybody know why they put n + 1 for 16kg?
If we do that, the problems becomes easy to solve.</p>
<p>Thanks,</p>
| 7,940 |
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<p>I've checked specs of Russian "Dynamic Albedo of Neutrons" module on Mars MSL rover - it can do 10^8 neutron pulses at 10 pulses per second. It works on D-T reaction in tiny linear accelerator, and power consumption is quite low (14W).</p>
<p>If you check specs for all these fusors (Farnsworth, Polywell, e.t.c) - they are happy to have 10^6 neutrons (best ones barely scratch 2*10^8) per second while having huge power consumption and short construction lifetime (<1hour). Does that mean that fusors are just a waste of paper from the science prospective, and real scientists routinely make neutron sources based on nuclear fusion with much better parameters/efficiency?</p>
<p><strong>Update:</strong>
Fusor: <a href="http://en.wikipedia.org/wiki/Fusor" rel="nofollow">http://en.wikipedia.org/wiki/Fusor</a></p>
<p><img src="http://i.stack.imgur.com/LyXIY.jpg" alt="enter image description here"></p>
<p>DAN: <a href="http://mars.jpl.nasa.gov/msl/mission/instruments/radiationdetectors/dan/" rel="nofollow">http://mars.jpl.nasa.gov/msl/mission/instruments/radiationdetectors/dan/</a>
More info on Russian: <a href="http://www.federalspace.ru/main.php?id=58" rel="nofollow">http://www.federalspace.ru/main.php?id=58</a> </p>
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] |
<p>Anecdotal evidence has it that a bottle of soda that was heavily shaken will not bubble over if tapped at the side multiple times.</p>
<p>Yet I wonder: Has the tapping really any effect? Or could it be that the mere time that passes while tapping at the side of the bottle has the carbon dioxide not mixed with the soda anymore?</p>
| 7,942 |
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<p>I found an equation in theory about magnetic induction in a <a href="http://en.wikipedia.org/wiki/Solenoid" rel="nofollow">solenoid</a>: $B_s=\mu_0 I n$. It should be magnetic induction for infinite length solenoid. I wonder if this is anyhow useful. Where can this be used? </p>
<p>($n = \frac {N}{L} $, where $L$ is length of solenoid and $N$ is number of turns... which doesn't make sense to me, if the length is supposed to be infinite)</p>
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<p>Using the formula $F=G\frac{m_1m_2}{d^2}$ where $m_1$and $m_2$ are the masses of two objects, $G$ is the gravitational constant, and $d$ is the distance between the objects, it is possible to calculate the force of the gravitational attraction between the objects in Newtons. However, since light is affected by black holes, the property required to interact gravitationally is energy, not mass.</p>
<p>How can I calculate the gravitational attraction between objects using their energies instead of their masses?</p>
| 7,944 |
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<p>From what I hear, some modern mathematical approach quantum field theory uses the following definition</p>
<blockquote>
<p>"A $d$-dimensional $S$-structured quantum field theory $Q$ is a mathematical object, consisting of its partition function $Z_Q$, its space of states $H_Q$, and its submanifold
operators $V_Q$, satisfying various axioms." (taken from some lecture notes)</p>
</blockquote>
<p>In physics textbooks, there is a definition of the partition function $Z$ in terms of an classical field action (much like the <a href="http://en.wikipedia.org/wiki/Partition_function_%28quantum_field_theory%29" rel="nofollow">wikipedia</a> entry).</p>
<p>I would like to know if there is a way to recover the partition function of a <a href="http://en.wikipedia.org/wiki/Wightman_axioms" rel="nofollow">Wightman</a> QFT (theory given in terms of fields operator valued distributions) or a <a href="http://en.wikipedia.org/wiki/Haag-Kastler_axioms" rel="nofollow">Haag-Kastler</a> QFT (given in terms of a local net of operator algebras).
I ask this because, from what I understand, those need not to come from a classical field theory derived from an action principle.</p>
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] |
<p>Are <a href="http://en.wikipedia.org/wiki/Soliton" rel="nofollow">solitons</a> an example of collective motion? </p>
| 7,946 |
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<p>I have a simple static mechanical system, but I reach a conclusion that seems to me counter-intuitive:</p>
<p><img src="http://i.stack.imgur.com/ueJV4.png" alt="enter image description here"></p>
<p>There is a pulley fixed to the ceiling and there is a weight fixed to a rope which goes through the pulley and is fixed to a point on the floor. I denote by $T$ the tension in the rope and draw the forces applied to the weight (since the weight is in equilibrium, I should have $T=mg$), and to the pulley. Since the pulley is in equilibrium, the force exerted on it by the ceiling should be $2T$, and thus I arrive at the conclusion that in this setting, the force exerted on the ceiling (by Newton's 3rd law) is equal to $2mg$. Is my reasoning correct?</p>
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<p><strong>Update:</strong> Trimok and MBN helped me solve most of my confusion. However, there is still an extra term $-(2/r)T$ in the final result. Brown doesn't write this term, and it seems physically wrong.</p>
<p><strong>Update #2:</strong> Possible resolution of the remaining issue. See comment on MBN's answer.</p>
<p>Suppose we have a rope hanging statically in a Schwarzschild spacetime. It has constant mass per unit length $\mu$, and we want to find the varying tension $T$. Brown 2012 gives a slightly more general treatment of this, which I'm having trouble understanding. Recapitulating Brown's equations (3)-(5) and specializing them to this situation, I have in Schwarzschild coordinates $(t,r,\theta,\phi)$, with signature $-+++$, the metric</p>
<p>$$ ds^2=-f^2 dt^2+f^{-2}dr^2+... \qquad , \text{ where} f=(1-2M/r)^{1/2} $$</p>
<p>and the stress-energy tensor</p>
<p>$$ T^\kappa_\nu=(4\pi r^2)^{-1}\operatorname{diag}(-\mu,-T,0,0) \qquad .$$</p>
<p>He says the equation of equilibrium is:</p>
<p>$$ \nabla_\kappa T^\kappa_r=0 $$</p>
<p>He then says that if you crank the math, the equation of equilibrium becomes something that in my special case is equivalent to</p>
<p>$$ T'+(f'/f)(T-\mu)=0 \qquad ,$$</p>
<p>where the primes are derivatives with respect to $r$. This makes sense because in flat spacetime, $f'=0$, and $T$ is a constant. The Newtonian limit also makes sense, because $f'$ is the gravitational field, and $T-\mu\rightarrow -\mu$.</p>
<p>There are at least two things I don't understand here.</p>
<p>First, isn't his equation of equilibrium simply a statement of conservation of energy-momentum, which would be valid regardless of whether the rope was in equilibrium?</p>
<p>Second, I don't understand how he gets the final differential equation for $T$. Since the upper-lower-index stress-energy tensor is diagonal, the only term in the equation of equilibrium is $\nabla_r T^r_r=0$, which means $\mu$ can't come in. Also, if I write out the covariant derivative in terms of the partial derivative and Christoffel symbols (the relevant one being $\Gamma^r_{rr}=-m/r(r-2m)$), the two Christoffel-symbol terms cancel, so I get</p>
<p>$$ \nabla_r T^r_r = \partial _r T^r_r + \Gamma^r_{rr} T^r_r - \Gamma^r_{rr} T^r_r \qquad , $$</p>
<p>which doesn't involve $f$ and is obviously wrong if I set it equal to 0.</p>
<p>What am I misunderstanding here?</p>
<p><em>References</em></p>
<p>Brown, "Tensile Strength and the Mining of Black Holes," <a href="http://arxiv.org/abs/1207.3342" rel="nofollow">http://arxiv.org/abs/1207.3342</a></p>
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<p><strong>Preface:</strong></p>
<p>I'm currently sitting at my desk on a 5th floor in a South Florida office building, as I was earlier this morning when I felt the building sway slightly. It wasn't continuous and the building swayed slightly one way and stopped. The sway was so subtle that others in the office didn't notice it (either walking or moving at their desk). Though, others did notice, corroborating my observations.</p>
<p><em>Additionally, I've spoken with more of my peers and they've stated that the building has swayed stronger than what I experienced this morning.</em></p>
<p><strong>Questions:</strong></p>
<p>I 'm somewhat concerned regarding the overall integrity of the building, its foundation and whether or not this is normal.</p>
<ul>
<li>Is it normal for a 6 story building to sway, in Florida on a calm day?</li>
<li>What could be causing this effect?</li>
<li>What steps should I take in order to get the issue/issues addressed?</li>
<li>Should I be concerned?</li>
</ul>
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<p>I remember many years ago, I think at 8th grade, seeing the teacher show us a <a href="https://en.wikipedia.org/wiki/Crookes_radiometer" rel="nofollow">Crookes radiometer</a>. I remember it being very fascinating. Today I read the wiki article on it, after looking up what it was called, but the article wasn't very clear in my opinion. Essentially the molecules that hit the dark sides have more energy and thus exert pressure which causes the device to rotate. And something about Einstein...</p>
<p>Among the lacks in the article is that it didn't clarify how effective this device is. It seems to me that relatively little energy input results a rather striking output (at least subjectively). So if I heat such a device with, say, $100 \ W$, how many watts would the output be in comparison?</p>
<p>Could such a device be built in large scale and thus used to convert solar energy to movement?</p>
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<p>Gamma radiation follows the inverse square law, I understand this as "double the distance, quarter the intensity"</p>
<p>So if you have a gamma source, at the source (distance = 0), the Intensity is $I_0$, and say at distance = 1, the Intensity is $\frac{I_0}{2}$ (You can't work this out just from the fact it follows the inverse square law right? You'd need the constant?)</p>
<p>So at distance = 2, while the intensity be a quarter of the original intensity so $\frac{I_0}{4}$ or a quarter of the intensity at the distance(1) that was doubled, so $\frac{I_0}{8}$?</p>
<p>I ask because I think this graph, which shows intensity of gamma radiation vs distance according the inverse square law, is wrong?</p>
<p><img src="http://i.stack.imgur.com/bwkyW.jpg" alt="enter image description here"></p>
<p>(also I don't see how it gets from 3x to $\frac{I_0}{8}$ because $3^2=9$)</p>
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] |
<p>After this semester, I'll have a background up to a first course in QFT (first 5 or 6 chapters of Peskin and Schroeder). </p>
<p>The next step in QFT will be something specific to the Standard Model (Elementary particles, QCD, etc).</p>
<p>Is that needed before going to study String Theory or not?</p>
| 7,952 |
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<p>How is it proved to be always true? It's a fundamental principle in Physics, that is based on all of our currents observations of multiple systems in the universe, is it always true to all systems? Because we haven't tested or observed them all.
Could it possible that we discover/create a system that could lead to a different result?</p>
<p>How are 100% sure that energy is always conserved?
Finally, why did we conclude it's always conversed? What if a system keeps doing work over and over and over with time? </p>
| 307 |
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] |
<p>I am reading about the Euler Equations of Fluid dynamics from
Leveque's numerical methods for conservation laws.</p>
<p>After introducing the mass, momentum and energy equations, some thermodynamic
concepts are discussed, to introduce an equation of state. </p>
<p>He says </p>
<pre><code>In the euler equations we assume that the gas is in chemical and thermodynamic
equilibrium and that the internal energy is a known function of pressure and density.
</code></pre>
<p>After this , the usual thermodynamics-related EOS discussions are carried out.</p>
<p>Now chemical equilibrium I understand (number of moles of the chemical constituents do not change), however I don't understand how the assumption
of thermodynamic equilibrium can be imposed. </p>
<p>From what baby thermodynamics I know, any thermodynamic analysis is always
calculated for quasi-static processes, like 'slowly' pushing a piston
in a cylinder of gas. </p>
<p>But in fluid <em>dynamics</em> fluids are <strong>flowing</strong> and that too rapidly and from intuition
there will not be any thermodynamic equlibrium during fluid flow. </p>
<p>Where is my understanding going wrong? </p>
| 7,953 |
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] |
<p>How does one get the value of acceleration of gravitation on earth accurately to 5 significant digits by experiment without electronic device?</p>
| 7,954 |
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] |
<p>I was shocked while reading Kittel's "Introduction to Solid State Physics", that the solid state of noble gases is a well described and makes one of the fundamental achievements of solid state physics.</p>
<p>So I wonder: is every possible element and every possible compound able to exist in any of the 3 states of matter?</p>
| 7,955 |
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<p>If dark energy is everywhere around us, then why don't we get separated? For example why don't I get separated from the pen kept in front of me? Or take a similar example in free space. Is dark energy's power greater than gravitational power?</p>
| 7,956 |
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<p>Undoubtedly, people use a variety of programs to draw diagrams for physics, but I am not familiar with many of them. I usually hand-draw things in <a href="http://www.gimp.org/">GIMP</a> which is powerful in some regards, but it is time consuming to do things like draw circles or arrows because I make them from more primitive tools. It is also difficult to be precise. </p>
<p>I know some people use LaTeX, but I am not quite sure how versatile or easy it is. The only other tools I know are Microsoft Paint and the tools built into Microsoft Office.</p>
<p>So, which tools are commonly used by physicists? What are their good and bad points (features, ease of use, portability, etc.)?</p>
<p>I am looking for a tool with high flexibility and minimal learning curve/development time. While I would like to hand-draw and drag-and-drop pre-made shapes, I also want to specify the exact locations of curves and shapes with equations when I need better precision. Moreover, minimal programming functionality would be nice additional feature (i.e. the ability to run through a loop that draws a series of lines with a varying parameter).</p>
<p>Please recommend few pieces of softwares if they are good for different situations.</p>
| 796 |
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<p>I'm running an experiment -- for the question, it doesn't matter which one, but I'm measuring an optical intensity $I$ as a function of two parameters: reflection angle $\theta$ and wavelength $\lambda$. I have motion control in place to move the setup to an angle $\theta_0$, and then I measure $I(\theta_0, \lambda)$ all at once using a spectrometer. I then move to the next angle $\theta_1$ and repeat.</p>
<p>Due to the beam travelling through different media at different angles, I have to move the position of the detector $p$ slightly for each angle, which is also automated. I should move the detector so that $M(p) = \sum_\lambda I(\theta_n, \lambda, p)$ is maximized. (M stands for "figure of Merit".)</p>
<p>$M(p)$ approximately has the form of a Gaussian plus noise, but I should be able to maximize it without caring what form it has. I have an amateurish algorithm in place to search for the proper position $p$ in order to maximize $M(p)$.</p>
<p>The quick-n-dirty algorithm steps $p$ in one direction by a step size $\Delta p$, until the value of $M(p)$ becomes smaller than a previous value. Then it goes back one step and tries a smaller step size $\Delta p$ in the other direction. As you can see, it doesn't account for measurement noise. My thoughts on how to improve it were in the direction of measuring a small number of points spaced $\Delta p$ apart and then fitting a parabola through them.</p>
<p>My question is, before I sit down and design a better algorithm, can anyone suggest an already-existing algorithm? I don't think a feedback algorithm (such as PID control) is appropriate, since I'm not trying to maintain a certain setpoint under perturbation of the system -- I just need to optimize to one value for each measurement. For bonus points, can someone point me to some papers on this subject?</p>
| 7,957 |
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<p>As $E=hf=\frac{hc}{\lambda}$, blue light - with a smaller wavelength - should have a higher energy. However, it is the case that blue light scatters the most. Why is it that higher energy rays scatter more?</p>
| 7,958 |
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<p>I saw the question <a href="http://physics.stackexchange.com/questions/3038/what-are-field-quanta">What are field quanta?</a> but it's a bit advanced for me and probably for some people who will search for this question.</p>
<p>I learned QM but not QFT, but I still hear all the time that "particles are the quanta of fields" and I don't really understand what it means.</p>
<p>Is there a simple explanation for people who know QM but not QFT?</p>
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<p>The question is pretty clear I think.</p>
<p>I only started learning GR but the results I've seen so far and
from some lectures I saw from Leonard Susskind on YouTube I understood that for low energy densities the curvature of space-time is mainly in the time coordinate.</p>
<p>Is this true, is the gravity we feel here on earth mainly the result of curvature of the time coordinate?</p>
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] |
<p>Could anyone explain the following expression: Why can mass not be considered concentrated at CM (center of mass) for rotational motion?</p>
| 7,961 |
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<p>I have a typical single-plaquette partition function for a gauge-field
$$ Z=\int [d U_{\text{link}}] \exp[-\sum_{p} S_{p}(U,a)]$$
with $U$ as the product of the the $U$'s assigned to each link around a plaquette. Now the $U$'s are irreducible representations of my group elements, which in my case is SU(2), and lets take the 1/2 representation as an example, then define the character as $\Xi_{r}\equiv \text{Tr}[U]$ which for our case is $\Xi_{1/2}=\text{Tr}[U]$. Now I have to take the product of these representations, however (and here's my question), how do I know which group elements/that-element's-representation to assign to each link? </p>
<p>I'm not sure how to compute the Trace without knowing first how to do the product of the link's representations, but I don't even know how to assign the elements to the links.</p>
<p>Thanks,</p>
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] |
<p>The Einstein-Hilbert action of general relativity is uniquely determined by general covariance and the requirement that only second derivatives in the metric appear. Yang-Mills theory can be motivated in a similiar way. In the original paper of Scherk, Julia, Cremer there are some arguments given from which they deduced the form of the action. They are only sketched however. Is there a more complete exposition of the derivation in the literature, or possibly even a uniqueness result as in the case of general relativity or Yang-Mills theory?</p>
| 7,963 |
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<p>I understand this topic well enough to get all the task done because they aren't very creative. But for my exam I think I should have this clear. </p>
<p>During the acceleration the force from the engine is of course bigger than air resistance and friction. This force, can we find it? And then the entire force the engine applies for the acceleration. Not just the stub you after subtracting for friction and air resistance. </p>
<p>$W = F \times s$<br>
$F = m \times a$</p>
<p>We have all the work done by forces at work, and the stretch of road is easy to calculate. If I now do this $\frac{F}{m} =a $ will that output be the correct acceleration? And this force that we found, is that a sum force? Because if that's the force sum on the car I can't find the engines power output which is what I want.
And what about the friction force at work, I think we can't find it when we just have the change in kinetic energy. Is that right? Primarily I would like to know if the change in kinetic energy can be tied somehow to the engines output during the acceleration. </p>
<p>The book has this nice equation too: $P = F \times v $<br>
But that's just constant speed. </p>
<p>Since we know the time maybe this can be used: $P = \frac{W}{T}$<br>
That just seems a little too easy. </p>
<p>Edit: I got a B on the exam, which means I'll be at again this fall. Not due to this question. Haha. </p>
| 7,964 |
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] |
<p>In this <a href="http://brokenspines.wordpress.com/2010/03/12/the-golden-ratio-1-61803399/">blog post</a>, I found this picture:</p>
<p><img src="http://i.stack.imgur.com/Qi3C2.jpg" alt="enter image description here"></p>
<p>Does the water really form golden ratio spiral in such cases? Or is the photo just a provocative example, without physics grounds for claims about "goldness" of the spiral?</p>
| 7,965 |
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<p>I have a charcoal smoker that uses water. My understanding is that the water serves as a buffer and as a way to add moisture to the cooking environment. Some say that using water wastes fuel because you have to heat the water. </p>
<p>If I am applying Newton's Conservation Law correctly, it seems to me that the heat put into the water is also released from the water as heat energy which also heats the food. So, I don't see that as wasteful. Is more energy lost in that part of the process than if no water was used?</p>
<p>It seems to me that the only waste is the heat left in the water after you are finished cooking. If you start with hot water from the tap, there's even less heat wasted there. </p>
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<p>I don't understand a particular statement in the QFT book by Klauber. The particular page I'm having difficulty on is page 67 of chapter 3 (<a href="http://www.quantumfieldtheory.info/website_Chap03.pdf" rel="nofollow">PDF link</a>).</p>
<p>The big picture is that the author wishes to investigate what the (operator) solutions to the Klein-Gordon equation, $\phi(x)$ and $\phi^\dagger(x)$, do when acting on the vacuum state $|0\rangle$. As prep for this, he creates a "general single particle state" ("general" meaning non $\mathbf{k}$-eigenstate) by operating on the vacuum with the operator
$$C\equiv\sum_\mathbf{k}A_\mathbf{k}a_\mathbf{k}^\dagger,\tag{3-108}$$
$$C|0\rangle=\sum_\mathbf{k}A_\mathbf{k}a_\mathbf{k}^\dagger|0\rangle=A_1|\phi_1\rangle+A_2|\phi_2\rangle+\cdots\equiv|\phi\rangle\tag{3-109}$$
Each $A_\mathbf{k}$ is just a number, the absolute value square of which represents the probability of finding the $\mathbf{k}$ eigenstate for the single particle.</p>
<p>The new state $C|0\rangle=|\phi\rangle$ is interpreted as a single particle state in a superposition of $\mathbf{k}$-eigenstates $|\phi_k\rangle$. The subscript $\mathbf{k}$ represents different momenta.</p>
<p>For probability/normalization arguments, the numbers $A_\mathbf{k}$ should obey
$$\sum_\mathbf{k}\left|A_\mathbf{k}\right|^2=1.\tag{3-110}$$
I feel like I understand the above statements.</p>
<p>The author then introduces the "general single particle destruction operator"
$$D\equiv\sum_\mathbf{k}a_k,\tag{3-111}$$
and shows that when applied to our general single particle state $|\phi\rangle$ above, the vacuum is (re)produced:
$$\begin{eqnarray}
D|\phi\rangle&=&\left(\sum_\mathbf{k}a_k\right)A_1|\phi_1\rangle+\left(\sum_\mathbf{k}a_k\right)A_2|\phi_2\rangle+\cdots\\
&=&A_1\underbrace{a_1|\phi_1\rangle}_{=|0\rangle}+A_1\underbrace{a_2|\phi_1\rangle}_{=0}+A_1\underbrace{a_3|\phi_1\rangle}_{=0}+\cdots+\\
&\ &+A_2\underbrace{a_1|\phi_2\rangle}_{=0}+A_2\underbrace{a_2|\phi_2\rangle}_{=|0\rangle}+ A_2\underbrace{a_3|\phi_2\rangle}_{=0}+\cdots+\\
&\ &+\cdots\\
&=&\underbrace{\left(A_1 + A_2 + \cdots\right)}_\text{can normalize = 1}|0\rangle.
\end{eqnarray}\tag{3-112}$$
(Note the subtle but important differences in the underbraces; some are $0$ while others are $|0\rangle$.)</p>
<p><strong>The part I am struggling with</strong> is understanding how the underbrace "can normalize = 1" at the end of $\text{(3-112)}$ can be true given $\text{(3-110)}$. It seems to me that the $A$ terms appearing at the end of $\text{(3-112)}$ are the same ones defined in the construction operator $C$ and normalized so that their absolute values squared sum to $1$. How can their just-plain sum also be of magnitude $1$? I know that one would *like * the underbraced term to sum to zero, but I don't see how that can be.</p>
<hr/>
<p>It was suggested I consider the quantity $\langle\phi|D^\dagger D|\phi\rangle$. Here is my attempt to calculate it.</p>
<p>$$
\begin{eqnarray}
\langle\phi|D^\dagger D|\phi\rangle&=&\langle0|(A_1^\dagger+A_2^\dagger+\cdots)(A_1+A_2+\cdots)|0\rangle=\langle0|\sum_\mathbf{j}\sum_\mathbf{k}A_\mathbf{j}^\dagger A_\mathbf{k}|0\rangle\\
&=&\sum_\mathbf{j}\sum_\mathbf{k}A_\mathbf{j}^\dagger A_\mathbf{k}\underbrace{\langle0|0\rangle}_{=1}=\underbrace{\sum_\mathbf{j}\sum_\mathbf{k}A_\mathbf{j}^\dagger A_\mathbf{k}}_\text{Can't simplify}\ne1
\end{eqnarray}
$$</p>
| 7,967 |
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-0.00358512531965971,
-0.008772515691816807,
-0.01599254086613655,
0.004568821284919977,
0.03778998181223869,
0.010135108605027199,
-0.011294378899037838,
0.00395465362817049,
0.009883123449981213,
-0.010948793031275272,
0.0016043753130361438
] |
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