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<p>I am trying to simulate liquid film evaporation with free boundary conditions (in cartesian coordinates) and my boundary conditions are thus:
$$
\frac{\partial h}{\partial x} = 0, \qquad (1)
$$
$$
\frac{\partial^2 h}{\partial x^2} = 0, \qquad (2)
$$
$$
\frac{\partial^3 h}{\partial x^3}=0. \qquad (3)
$$
However, I need only two of the above three conditions to satisfy my 4th order non-linear partial differential equation for film thickness, which looks something like.
$$
\frac{\partial h}{\partial t} + h^3\frac{\partial^3 h}{\partial x^3} + ... = 0
$$
My question is: what does a combination of 1st and 2nd derivative conditions mean and what does a combination of 2nd and 3rd derivatives mean?</p>
<p>If I apply (1) and (2), does it mean that slope and curvature are zero and if I apply (1) and (3), does it mean that slope and shear stress are zero (from analogies of bending beams etc.)</p>
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<p>Sometimes I volunteer to help freshmen and sophomore students with introductory physics and calculus courses. The problem is most of them are not stimulated and take these courses because they are obligatory.</p>
<p>On the other hand some TV series, such as CSI Miami, contains scenes which show how some of the technologies that based on physics principles can be used in some jobs for example to help solving a crime. When the young generation see such TV shows they can actually get stimulated about science and physics and it can show them that it is not only useful but also it can open to them real exciting career opportunities. Of course the scientific content can be misleading.</p>
<p>Because of that, I was wondering if there is(are) some site(s) which discuss the science of such TV series and other series to filter the true science from the exaggeration. One can use such "filtered" interesting examples to stimulate freshmen and sophomore students towards science in general and physics in particular and may be math.</p>
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<blockquote>
<p><strong>Possible Duplicate:</strong><br>
<a href="http://physics.stackexchange.com/questions/7470/chemical-potential">Chemical potential</a> </p>
</blockquote>
<p>The chemical potential of electron and positron is equal but with opposite sign. How one can visualize the negative chemical potential of positron, physically?</p>
| 239 |
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<p>I am familiar with the true (or general) notion of orthogonality, given in the Linear Algebra and <a href="http://math.stackexchange.com/a/11516/39930">Pythagoras theorem derived from the $\vec x \cdot \vec y = 0$</a>. I have also recently got to know that true or general definition of orthogonality is that orthogonal things are mutually exclusive and spin "up" is orthogonal to spin "down". </p>
<p>I always believed that two elements are orthogonal when measuring one component does not give any information about the other. It is therefore paradoxical to me to hear that spin up state is orthogonal to spin down. Having x and y in opposite directions implies that if you measure $x=n$, you are sure that $y=-n$. They have perfect overlap/correlation. Despite we agree with physicians that orthogonality is opposite of "overlap", they say that up and down, which perfectly overlap, are orthogonal! Orthogonality is identified with its opposite. I cannot screw my brain around this. </p>
<p>The last time I heard this idea was in Susskind's Theoretical Minimum, lecture 2, where he recalls that "overlap/correlation is opposite of orthogonality and orthogonality means mutual exclusiveness so that you can clearly distinguish between orthogonal things" (no overlap between basis vectors). Why does he speak about "measurably distinct"? Which measurement in conventional space does distinguish between x and y?</p>
<p>I don't understand why spin down is orthogonal to spin up rather than left and right are orthogonal to spin up. Does this orthogonality have anything to do with the conventional, 90° angle, like up-left, orthogonality? What does Euclidian orthogonality of 90° have in common with mutual exclusiveness?</p>
<p>This question has another dimension. Susskind comes up with 3 orthogonal basises. He says that we can have up-down orthogonality, but, we can also have left-right orthogonality. I want to know how these orthogonalities are related? If spin up and spin down are orthogonal along z axis, then what is relationship between $x$ and $z$? Why there are only 3 such relationships? <a href="http://www.youtube.com/watch?feature=player_detailpage&v=a6ANMKRBjA8#t=3351s" rel="nofollow">Here is how Susskind derives the "left-right" orthogonality relationship</a> from the up and down relationship. </p>
<p>In short, I want to know what is common between all kinds of orthogonalities, what makes <em>up</em> exclusive with <em>down</em>, aren't up and down 100% correlated/overlapped, why they are not exclusive/orthogonal in the ordinary space and what is the relationship between between up-down and left-right bases?</p>
<p>In math you can define 3 curls, the vectors of rotation. Can you rotate a thing in all 3 planes simultaneously or 3rd rotation will be a combination of (degree of freedom is two). Also, I see that </p>
<p>$${left - right \over \sqrt 2} = up$$
$${left + right \over \sqrt 2} = down$$</p>
<p>Is this because of the same orthogonality between sine, cos and complex exponential?</p>
<p>There is <a href="http://stats.stackexchange.com/questions/12128/what-does-orthogonal-mean-in-the-context-of-statistics">a question</a> asking how statistical independence is related to orthogonality. Can we say which properties are common between statistical independence and orthogonality? This can be interesting particularly in the context of QM, which is sorta statistical mechanics.</p>
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<p>Based on my understanding of physics after seeing <em>The Distinction of Past and Future</em> on <a href="http://research.microsoft.com/apps/tools/tuva/#data=3%7C4dbfe549-e795-47a0-bda2-9597fe5bb344%7C%7C" rel="nofollow">Project Tuva</a>, there is no distinction between past and future on a fundamental level- all particle interactions can occur in reverse. So my question is whether or not one could theoretically reverse the direction of all particles in the observable universe relative to each other and have time essentially go backwards indefinitely.</p>
<p>If you think about it, things could "fall" upwards because the air resistance would be much lower due to the way the air was moving when it fell, and the velocity from the gravity downwards would be reversed as well as air under the ball pushing up (again due to the way the air was moving previous to the switch). I don't see why this same logic couldn't be applied to a more complex system.</p>
<p>Does this logic make sense? If not, where is the flaw? What other constraints would need to be added to make time essentially go backwards other than reversing direction, if it is possible at all, in theory?</p>
| 6,942 |
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<p>What would be the rate of temperature loss for an average sized human in space without a suit? A human generates about 100 watt at rest. But how can we use that to calculate how fast the temperature will go down? Also how much heat would be absorbed by the sun? Assume the person is somewhere between the earth and the moon.</p>
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<p>Warning: I am not a physicist so please excuse my naivety!</p>
<p>As you all know, physicists think that there exist four fundamental forces.</p>
<p>Would a universe with zero fundamental forces be possible, at least in principle? Would elementary particles be able to exist in such a universe? On a larger scale, what would such a universe look like?</p>
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<p>Consider a spherical symmetric thin cell of photons converging to a point. At some moment, there is a formation of an horizon and a black hole. But each black hole is evaporating,and so, after some time, all the black hole has evaporated.</p>
<p>So, we could consider an "initial state", well before the creation of the horizon, and a "final state", well after the end of the evaporation of the black hole.</p>
<p>If we call $E$ the total energy of the converging photon cell,
what is the difference of entropy between the "initial state" and the "final state"?</p>
| 6,945 |
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<p>How to focus beam from the fresnel lens on a flat surface.</p>
<p><img src="http://i.stack.imgur.com/tINjl.png" alt="enter image description here"></p>
<p>In my case, instead of producing the light beam, I am receiving the light beam.
So, the beam comes from the right and then focuses at a focal point.</p>
<p>My problem is that I am using cylindrical fresnel lens</p>
<p><img src="http://i.stack.imgur.com/T6BPT.jpg" alt="enter image description here"></p>
<p>And light beam can come from any direction (360 deg). Therefore the focal point is in the middle of the cylinder. </p>
<p>I am sensing the beam, using photodiode. Since photodiode has a flat surface, if i will simply put it in the middle of the cylinder it will not absorb most of the energy of a beam. Therefore somehow I need to reflect the beam from the focal point in the center of a cylinder to the flat surface, say the bottom of a cylinder, where I could put my photodiode.</p>
<p>How can this be done?</p>
| 6,946 |
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] |
<p>I'm trying to find the relation between current density $\boldsymbol{j}$, voltage $v$, and magnetic field $\boldsymbol{B}$ for the time-harmonic approximation in a cylindrically-symmetric coil (a torus if you will) onto which a current is applied. The coordinates are $r$, $z$, $\theta$.</p>
<p>What I got so far is:</p>
<ul>
<li>The current density can be written as
$$
\boldsymbol{j} = \exp(\text{i}\omega t) j \boldsymbol{e}_{\theta}
$$
with a scalar function $j$.</li>
<li>The magnetic vector potential $\boldsymbol{A}$ if the magnetic field $\boldsymbol{B}$($=\nabla\times \boldsymbol{A}$) has the form
$$
\boldsymbol{A} = \exp(\text{i}\omega t) \phi \boldsymbol{e}_{\theta}
$$
with a scalar function $\phi$ (and the angular frequency $\omega$); $\boldsymbol{e}_\theta$ is the unit vector in direction $\theta$.</li>
</ul>
<p>From Maxwell's equations, it follows that
$$
\nabla\times \boldsymbol{E} + \text{i}\omega \boldsymbol{B} = \boldsymbol{0},
$$
so
$$
\nabla\times (\boldsymbol{E} + \text{i}\omega \boldsymbol{A}) = \boldsymbol{0}.
$$
With Ohm's law, $\boldsymbol{j} = \sigma \boldsymbol{E}$, and the above representations, we have
$$
\nabla\times ((\sigma^{-1} j + \text{i}\omega \phi)\boldsymbol{e}_{\theta}) = \boldsymbol{0}.
$$
This means that the quantity $r(\sigma^{-1} j + \text{i}\omega \phi)$ must be a constant in the coil. I suspect that it takes a value that depends on the applied voltage $v$, but I fail to see how that would come in.</p>
| 6,947 |
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<p>There was a Question bothering me.</p>
<p><img src="http://i.stack.imgur.com/qXmBT.png" alt="enter image description here"></p>
<p>I tried solving it But couldn't So I finally went up to my teacher asked him for help . He told me that there was a formula for Electrostatic pressure $\rightarrow$</p>
<p>$$\mbox{Pressure}= \frac{\sigma^2}{2\epsilon_0}$$</p>
<p>And we had just to multiply it to the projected area = $\pi r^2$ </p>
<p>When i asked him about the pressure thing he never replied. </p>
<p>So what is it actually.Can someone Derive it/Explain it please.</p>
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<p>Obviously in a simple, classroom style experiment the energy required to increase water temperature is a constant and thus wouldn't change, but what if we made the experiment just slightly more real world?</p>
<p>Imagine the following set up: a typical metal pot filled with a gallon of water with a heat source below it putting out a constant amount of energy. Now since the output of energy and the rate of absorption by the pot is constant (I assume) I think we can phrase the question as such:</p>
<p>Would the time it takes for the water to go from 50C to 60C be any different from the time it takes the water to go from 80C to 90C?</p>
| 6,949 |
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<p>I understand that gravity in GR is a manifestation spacetime curvature dictated by the field equations by the principle that objects follow the geodesic path in spacetime. </p>
<p>And, I get how gravitational radiation travels like waves because there are wave equations in the field equations.</p>
<p>But, I have trouble reconciling gravitational waves with larger scale spacetime curvature.</p>
<p>If the earth suddenly disappeared, then the metric field would begin to flatten out and become a Minkowski Spacetime and this effect would travel away from where the earth is, presumably by gravitational waves.</p>
<p>Likewise, if a planet suddenly sprang into existence, a Schwarzchild Spacetime would emerge.</p>
<p>What I have trouble understanding and visualizing is how this effect is propagated and sustained by gravitational waves. </p>
<p>Should I picture the gravitational waves as constantly acting in the field to produce curvature of spacetime? If so, how do the gravitational waves around us in Schwarzchild spacetime differ from the gravitational waves in other spacetimes to produce another metric? </p>
<p>Or am I thinking about this the wrong way?</p>
<p>Would it be fruitful to attempt to understand this by analogy with the propagation of Electromagnetic fields?</p>
| 6,950 |
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<p>Apparently <a href="http://en.wikipedia.org/wiki/Huygens%E2%80%93Fresnel_principle">Huygens' principle</a> is only valid in an odd number of spatial dimensions:</p>
<ul>
<li><p><a href="http://mathoverflow.net/a/5396/21349">http://mathoverflow.net/a/5396/21349</a></p></li>
<li><p><a href="http://physics.stackexchange.com/questions/129215/huygens-principle-in-curved-spacetimes">Huygen's principle in curved spacetimes</a></p></li>
</ul>
<p>Why is this?</p>
<p>[EDIT] This is somewhat perplexing, since AFAIK it's pretty common to teach freshmen about double- and single-slit diffraction using a two-dimensional analysis and invoking Huygens' principle. Does this work only because there's an ignored third axis of translational symmetry?</p>
<p>I wonder if it's possible to gain insight by making a grid and doing sort of a finite-element analysis.</p>
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<p>I have a bit of an understanding issue why the representations of $SO(3)$ are so important for Quantum Mechanics. When looking at its Irreps one gets the Spin and Angular Momentum operators and thus their physical significance is fairly obvious.</p>
<p>What I don't get is WHY these Irreps give what they give. When introducing the angular momentum operator by canonical quantization $\vec{L} = \vec{r} \times \vec{p}$ where $\vec{r}$ and $\vec{p}$ are operators one arrives at an operator whose entries satisfy the commutation relationships that the images of the Irreps of $SO(3)$ satisfy as well (i.e. $[L_1, L_2] = iL_3$ (up to some $\hbar$s)). </p>
<p>Wigner's theorem then tells us that every single vector operator acting on states that are eigenstates of a rotationally invariant operator (if I understood the theorem correctly).</p>
<p>My main question now is: Is the declaration of $SO(3)$ as a symmetry group a postulate? Is it "obvious?" If yes, why? When we talked about symmetries in classical mechanics it was either a postulate (Noethers Theorem gives us a conserved quantity for the symmetry postulate that the laws of nature are the same everywhere and constant in time) or one could explicitly calculate it (i.e. that a Hamiltonian or Lagrangian is invariant under the act of certain operations like rotations). Which (if any) of that is it in the case of $SO(3)$ and Quantum Mechanics?</p>
<p>I hope someone can shed some light on this for me.</p>
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<p>It is generally assumed, from a person's perspective, that pushing a cart is more easier than pulling one. But why?<br>
Is there any difference in terms of force required to achieve the same amount of displacement?</p>
<p>Or is it merely human perception?</p>
<p>Why is it that almost all automobiles transfer torque to the back axle. But then, why do trains have engines in the front?</p>
| 746 |
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<p>If I'm moving through MY reference frame at the speed of light, isn't time still passing by normally for me? </p>
<p>Help me think about this fourth dimension- space-time. I want to intuitively understand it. I understand some thought-experiments (Einstein's train experiment, for example, and why events occur slower/later for people that are farther away or moving from the event because of the infinitesimal amounts of time it takes for light to travel to them). </p>
<p>Another question I could ask here is what caused you to understand space-time when you finally got it (if you didn't immediately get it). </p>
<p>Thanks!</p>
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<blockquote>
<p>A cockroach is crawling along the walls inside a cubical room that has an edge length of 3 m. If the cockroach starts from the back lower left hand corner of the cube and finishes at the front upper right hand corner, what is the magnitude of the <a href="http://en.wikipedia.org/wiki/Displacement_%28vector%29" rel="nofollow">displacement</a> of the cockroach?</p>
</blockquote>
<p>Simple question, but I don't understand the answer they provide. In my mind the cockroach has a displacement of 3m. Can someone verify or point my intuition in the right direction.</p>
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] |
<p>So the <a href="http://en.wikipedia.org/wiki/Electric_field" rel="nofollow">electric field</a> between two parallel plates is given by $E = V/d.$ How do you derive this? </p>
| 6,955 |
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<p>In Peskin & Schroeder chapter 19 page 656, where the axial current anomaly of massless 2D QED is discussed, the authors go from: $$ \bar\psi(x+\varepsilon/2)\Gamma(x)\psi(x-\varepsilon/2)\tag{19.25} $$
(where $\Gamma(x)$ is some operator)
to: $$ \bar\psi(x+\varepsilon/2)\Gamma(x)\psi(x-\varepsilon/2) \tag{19.27} $$
(where now the two Fermionic fields are contracted) to: $$ Tr\left[\Gamma(x)S_F(\varepsilon)\right] \tag{19.27} $$
(where $S_F(x)$ is the Fermion propagator between a spacetime interval $x$)</p>
<p>I really don't understand these transitions and would appreciate any help with how to do them.
In particular:</p>
<p>1) Is it implicitly understood (from the very beginning of this derivation) that the axial current is in fact time-ordered (so that we can employ Wick's theorem) and always assume it operates on the vacuum (so that normal ordered terms vanish)?</p>
<p>2) Why does the contraction of the two Fermion fields <em>over</em> the $\Gamma$ operator lead to a trace, as if we had a loop? </p>
| 6,956 |
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<blockquote>
<p><strong>Possible Duplicate:</strong><br>
<a href="http://physics.stackexchange.com/questions/5587/a-pendulum-clock-problem">A pendulum clock problem</a> </p>
</blockquote>
<p>There are two pendulums.</p>
<p>First pendulum consists of a rod of length L and flat heavy disk of radius R (R < L), disk is connected rigidly to the rod such that the plane of the disk is vertical. Mass of the rod is negligible compared to mass of the disk. Center of the disk is at distance L from the upper point of the rod.</p>
<p>Second pendulum has same rod of length L and same disk, but the disk is not connected rigidly to the rod. Disk can rotate freely around its center at which it is attached to the rod at distance L from the upper end of the rod. Disk remains in vertical plane, as if there is ball bearing atached to the rod.</p>
<p>Which pendulum has smaller period at small deviations, if their periods are different.</p>
| 240 |
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<p>I'm questioning if there's a way to make use of the priciple of steam engines again. </p>
<p>The idea of a steam engine is a little primitive and outdated maybe, but the principle isn't that at all<br>
if you ask me.
In the 19th century, they used water for steam engines and I assume this fluid was chosen because of the wide availability and low costs.<br>
The cons of using water is that you have to burn fuel like coal for heat because of the relatively high boiling point of water.
Back in those days, those fuels were still widely available, but obviously those times have changed. </p>
<p>The thing is, if you use a fluid with a much lower boiling point you'd have to add only a little bit of energy onto it to make it boil.
So to obtain the best efficiency, you'd take a fluid which has a boiling point near the environment temperature.
For example, pentane boils at 309K / 36C. (I don't know which substance is more suitable)</p>
<p>I don't know how much energy you gain when this substance boils, but if you have a closed circuit for the fluid and a radiator, you could in theory build an engine which runs indefinitely.
Because the energy you gain from the boiling substance is big enough to generate the little heat which is needed to make it boil.
Maybe this is a stupid question, and i suppose this has been considerated a long time ago.<br>
But i'm just curious about it. And then again, the simple or weird ideas are the ones who actually work. :p
Thanks for any response.</p>
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<blockquote>
<p><strong>Possible Duplicate:</strong><br>
<a href="http://physics.stackexchange.com/questions/7041/speed-of-gravity-and-speed-of-light">“Speed” of Gravity and Speed of Light</a> </p>
</blockquote>
<p>I'm wondering if gravitational waves have the same speed of light? They must if gravity is mediated by a graviton spin 2 massless boson. One can argue that since gravity is a long range force, then its boson must have zero mass. But neutrinos travel through the universe too and despite their tiny mass yet have non zero mass. So, I'm wondering if there is a deeper justification for believing that gravity's speed =c?</p>
| 158 |
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<p>I know how we define a vacuum in flat space QFT and also in a curved space QFT. But, can somebody tell me how do the choice of vacuum state in (say) the CFT side of AdS/CFT changes the choice of vacuum state in gravity side? Let me ask the other way. I mean if we pick a vacuum (say in bulk side, because it may not be unique), how does it reflect on the CFT vacuum (and vice versa)? So, my question is how does this choice reflect on both sides and how do we generally make the identification?</p>
<p>Thanks.</p>
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<p>Measurements of consecutive sites in a many body qudit system (e.q. a spin chain) can be interpreted as generating a probabilistic sequence of numbers $X_1 X_2 X_3 \ldots$, where $X_i\in \{0,1,\ldots,d-1 \}$.</p>
<p>Are there any studies on that approach, in particular - exploring predictability of such systems or constructing a Markov model of some order simulating it? </p>
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] |
<p>According to <a href="http://www.nature.com/news/proof-found-for-unifying-quantum-principle-1.9352" rel="nofollow">this</a> recent article in <em>Nature</em> magazine, John Cardy's <em>a</em>-theorem may have found a proof.</p>
<p>Question:</p>
<ol>
<li><p>What would the possible implications be in relation to further research in QFT? </p></li>
<li><p>Specifically, what types of QFT's would now be studied more closely?</p></li>
</ol>
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] |
<p>I could understand the derivation of the "bulk-to-boundary" propagators ($K$) for scalar fields in $AdS$ but the iterative definition of the "bulk-to-bulk" propagators is not clear to me. </p>
<p>On is using the notation that $K^{\Delta_i}(z,x;x')$ is the bulk-to-boundary propagator i.e it solves $(\Box -m^2)K^{\Delta_i}(z,x;x') = \delta (x-x')$ and it decays as $cz^{-\Delta _i}$ (for some constant $c$) for $z \rightarrow 0$. Specifically one has the expression, $K^{\Delta_i}(z,x;x') = c \frac {z^{\Delta _i}}{(z^2 + (x-x')^2)^{\Delta_i}}$</p>
<ul>
<li><p>Given that this $K$ is integrated with boundary fields at $x'$ to get a bulk field at $(z,x)$, I don't understand why this is called a bulk-to-boundary propagator. I would have thought that this is the "boundary-to-bulk" propagator! I would be glad if someone can explain this terminology. </p></li>
<li><p><em>Though the following equation is very intuitive, I am unable to find a derivation for this and I want to know the derivation for this more generalized expression which is written as</em>,</p></li>
</ul>
<p>$\phi_i(z,x) = \int d^Dx'K^{\Delta_i}(z,x;x')\phi^0_i(x') + b\int d^Dx' dz' \sqrt{-g}G^{\Delta_i}(z,x;z',x') \times$
$\int d^D x_1 \int d^D x_2 K^{\Delta_j}(z,x;x_1)K^{\Delta_k}(z,x;x_2)\phi^0_j(x_1) \phi^)_k(x_2) + ...$</p>
<p>where the "b" is as defined below in the action $S_{bulk}$, the fields with superscript of $^0$ are possibly the values of the fields at the boundary and $G^{\Delta_i}(z,x;z',x')$ - the "bulk-to-bulk" propagator is defined as the function such that,</p>
<p>$(\Box - m_i^2)G^{\Delta_i}(z,x;z',x') = \frac{1}{\sqrt{-g}} \delta(z-z')\delta^D(x-x')$</p>
<ul>
<li>Here what is the limiting value of this $G^{\Delta_i}(z,x;z',x')$ that justifies the subscript of $\Delta_i$. </li>
</ul>
<p>Also in this context one redefined $K(z,x;x')$ as,</p>
<p>$K(z,x;x') = lim _ {z' \rightarrow 0} \frac{1}{\sqrt{\gamma}} \vec{n}.\partial G(z,x;z',x')$
where $\gamma$ is the metric $g$ restricted to the boundary. </p>
<ul>
<li><p>How does one show that this definition of $K$ and the one given before are the same? (..though its very intuitive..) </p></li>
<li><p>I would also like to know if the above generalized expression is somehow tied to the following specific form of the Lagrangian, </p></li>
</ul>
<p>$S_{bulk} = \frac{1}{2} \int d^{D+1}x \sqrt{-g} \left [ \sum _{i=1}^3 \left\{ (\partial \phi)^2 + m^2 \phi_i^2 \right\} + b \phi_1\phi_2 \phi_3 \right ]$ </p>
<p>Is it necessary that for the above expression to be true one needs multiple fields/species? Isn't the equation below the italicized question a general expression for any scalar field theory in any space-time? </p>
<ul>
<li>Is there a general way to derive such propagator equations for lagrangians of fields which keep track of the behaviour at the boundary? </li>
</ul>
| 6,960 |
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<p>I'm reading through a <a href="http://www.vis.uni-stuttgart.de/~dachsbcn/download/rsm.pdf" rel="nofollow">computer graphics paper</a> and author says that the radiant intensity emitted by an infinitely small surface point $p$ with normal $ n $ into direction $ \omega $ is</p>
<p>$$
I_p(\omega) = \phi_p max(0,<n|\omega>)
$$</p>
<p>where $\omega_p$ is the radiant flux emitted by the surface, and $<a|b>$ denotes the dot product.</p>
<p>First question: Am I correct in assuming that this is only true for an infinitely small solid angle?</p>
<p>The author then claims that the irradiance due to this surface point $p$ arriving at another surface point is</p>
<p>$$
E_p(x,n) = I_p(x-x_p) \frac{ max(0,<n|x-x_p>) }{||x-x_p||^4} = \phi_p \frac{max(0,<n_p | x-x_p>) max(0,<n|x-x_p>) }{||x-x_p||^4}
$$</p>
<p>where $x_p$ is the location of the emitting surface point, $x$ is the location of the surface point where the light arrives, $n$ its normal, and $||x||$ is the norm (length) of $x$. The author doesn't mention it but I'm assuming he's always talking about infinitely small surface points, since no areas seem to be involved.</p>
<p>Second question: Where does $||x - x_p ||^4$ come from? I know that the radiant flux incident on a surface point due to a light source diminishes with the square of the distance from the surface point to the light source, but this in my mind only constitutes a term of $||x - x_p||^2$. Where does the 4 come from?</p>
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<p>I'm a retired police officer trying to learn classical mechanics on my own. I have gone through many links on the Internet including the classical mechanics quick reference textbooks from Physics Stack Exchange. But, I always have the same problem just as anyone trying to learn classical mechanics <em>on his/her own</em> has had the experience of "going down the Classical Mechanics Rabbit Hole".</p>
<blockquote>
<p>It turns out that only classical mechanics is the most difficult part of physics to learn on ones own. I had a friend who confirmed this by comparing how difficult it is to learn classical mechanics (including Lagrangian and Hamiltonian formulation) on his own with electrodynamics and general relativity. (Who are much much more difficult that all the field of CM)</p>
</blockquote>
<p>For example, suppose you come across the novel term vector space, and want to learn more about it. You look up various definitions, and they all refer to something called a field. So now you're off to learn what a field is, but it's the same story all over again: all the definitions you find refer to something called a group. Off to learn about what a group is. Ad infinitum. That's what I'm calling here "to go down the Math Rabbit Hole."</p>
<p>For example, I had lot of difficulties with the book "An Introduction to Mechanics" by <em>Daniel Kleppner</em>, <em>Robert J. Kolenkow,</em> which seemed according to many views to be an easy approach toward Newtonian and relativistic mechanics. The authors in general only and quickly pushes equations in my front without giving any reason for why a certain procedure is correct, and give no explanation on most of the things. I had then one choice: search on the net. But when I do, to search for a term X, I get to wikipedia page X, who give a definition that contains another term Y, where I click to understand the full meaning of term X, but who then contain another term Z, who redirects to... which leaves me with no understanding.</p>
<p>Another thing is that when I go here on Physics Stack Exchange, and when I see answers like: </p>
<ul>
<li><a href="http://physics.stackexchange.com/a/14752/">http://physics.stackexchange.com/a/14752/</a></li>
<li><a href="http://physics.stackexchange.com/a/67705/">http://physics.stackexchange.com/a/67705/</a></li>
<li><a href="http://physics.stackexchange.com/a/71093/">http://physics.stackexchange.com/a/71093/</a></li>
<li><a href="http://physics.stackexchange.com/a/64976/">http://physics.stackexchange.com/a/64976/</a></li>
<li>and many many others... (like an answer by David Z for a question that kinda looks like: 'Would a heavier object fall faster because they attract earth stronger', I have no idea where he found the equations he wrote down. Also in many applied physics questions and answers by Lubos Motl. And in some questions: like: 'Why Newton's third law apply to all inertial frame?' I have no idea about that even if I already learn a lot from Daniel's book.)</li>
</ul>
<p>I don't know where those guys got all that stuff. I feel like: Mechanics is not well organized. For example, in relativity we first learn about Galilean relativity, then special relativity then general relativity. Everything is in order and it makes of the understanding a lot smoother. (according to my friend) But in classical mechanics I don't know where to start or what to pick.</p>
<p>In Lagrangian and Hamiltonian mechanics book, it is even worse.</p>
<p>Result? I fail to correctly answer some basic questions like: what happens when a cup of water starts to melt? or even more easy physics questions.</p>
<p><strong>So I'm searching for a clear textbook that explains Newtonian mechanics well, then goes to special relativity, then to Lagrangian and Hamiltonian mechanics.</strong></p>
<p>My dream for the next years of my life is to understand mechanics: Newtonian, SR, Lagrangian and Hamiltonian. And to start writing a web page about explanations of different phenomena like John Baez this week on mathematical physics. And maybe to do research on problems in classical physics which would make of me the most happy man in the world.</p>
<p>Regards. Thanks for your understanding and time.
My situation is similar to <a href="http://math.stackexchange.com/q/617625">this guy</a></p>
<p>My background: I'm very old, so I forgot almost all the math/physics I've got in school, however, I've taken courses on Algebra, trigonometry and single variable calculus using KhanAcademy and some MIT videos. I've taken an MIT test on CalcI (just downloading the test online and verifying the solutions) and I scored 90%.</p>
| 98 |
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<p>As an exercise for myself, I have been working on rewriting the massive vector boson propagator (unitary gauge). I have run into a problem interpreting some of the terms that stick around when the propagator is rewritten this way. Here's what I have: I've taken the unitary gauge vector propagator</p>
<p>$$
D_{\mu\nu}(q) = \frac{i}{q^2 - M_W^2 + i \varepsilon} \left( -g_{\mu \nu} + \frac{q_\mu q_\nu}{M_W^2} \right)
$$</p>
<p>and projected it into its helicity components. The convention I am using is that</p>
<p>$$
\epsilon^1_\mu(\vec q) = (0,1,0,0) \\
\epsilon^2_\mu(\vec q) = (0,0,1,0) \\
\epsilon^0_\mu(\vec q) = \frac{1}{M}(|q|,0,0,E_q) \\
\epsilon^s_\mu(\vec q) = \frac{1}{M}(E_q,0,0,|q|)
$$
where $E_q = \sqrt{M^2+|q|^2}$. These are an orthonormal set, where</p>
<p>$$
\epsilon^{\lambda}_\mu \epsilon^{\lambda' \mu} = - \eta_{\lambda} \delta_{\lambda \lambda'}
$$
where $\eta_\lambda = 1$ for $\lambda = \pm,0$ and $-1$ for $\lambda = s$, the scalar polarization. So, it's easy to show that</p>
<p>$$
X_{\mu \nu} = \sum_{\lambda,\lambda'} X_{\lambda,\lambda'} \epsilon^{\lambda}_\mu \epsilon^{\lambda' }_{\nu} \Rightarrow X_{\mu \nu} \epsilon^{\lambda \mu} \epsilon^{\lambda' \nu} = \eta_\lambda \eta_{\lambda'} X_{\lambda \lambda'}
$$</p>
<p>In particular,
$$
- g_{\mu \nu} \Rightarrow g_{\lambda \lambda'} = \frac{\eta_\lambda' \delta_{\lambda \lambda'}}{\eta_{\lambda} \eta_{\lambda'}} = \frac{\delta_{\lambda \lambda'}}{\eta_{\lambda}} = \eta_{\lambda}\delta_{\lambda \lambda'}
$$
Where I've run into difficulty is breaking up the transverse term.</p>
<p>$$
q_\mu \epsilon^{1\mu}(\vec q) = 0 \\
q_\mu \epsilon^{2\mu}(\vec q) = 0 \\
q_\mu \epsilon^{0\mu}(\vec q) = \frac{|q|}{M}(q_0-E_q) \\
q_\mu \epsilon^{s\mu}(\vec q) = \frac{1}{M}(q_0 E_q - |q|^2)
$$</p>
<p>The last of these can be rewritten</p>
<p>$$
q_\mu \epsilon^{s\mu}(\vec q) = M + \frac{E_q}{M}(q_0-E_q)
$$</p>
<p>So, the helicity components of</p>
<p>$$
T_{\mu \nu} = \frac{q_\mu q_\nu}{M^2}
$$</p>
<p>are</p>
<p>$$
T_{1\lambda} = T_{\lambda1} = T_{2\lambda} = T_{\lambda2} = 0
$$</p>
<p>$$
T_{00} = \frac{|q|^2}{M^2} \left( \frac{q_0-E_q}{M} \right)^2
$$</p>
<p>$$
T_{ss} = 1 + \frac{q_0^2-E_q^2}{M^2} + \frac{|q|^2}{M^2} \left( \frac{q_0-E_q}{M} \right)^2
$$</p>
<p>$$
T_{0s} = T_{s0} = - \frac{|q|}{M} \left( \frac{q_0-E_q}{M} \right) - \frac{E_q|q|}{M^2}\left(\frac{q_0-E_q}{M}\right)^2
$$</p>
<p>The term equal to $1$ in the scalar polarization term cancels out the corresponding scalar polarization term $g_{ss}$. Furthermore, all of the terms proportional to $(q_0-E_q)^2$ I understand. They cancel out the pole in the propagator, since</p>
<p>$$
\frac{1}{q^2 - M^2 + i \epsilon} = \frac{1}{q_0^2 - E_q^2 + i \epsilon} \sim \frac{1}{q_0 - E_q + i \epsilon} \frac{1}{q_0 + E_q}
$$</p>
<p>and after canceling out the pole they retain a factor $q_0 - E_q$ which forces them to be 0 while on-shell. These are explicitly off-shell corrections. However, I'm not sure how to interpret the terms</p>
<p>$$
T_{ss} \ni \frac{q_0^2-E_q^2}{M^2} = \frac{q_0+E_q}{M} \frac{q_0-E_q}{M}
$$</p>
<p>and</p>
<p>$$
T_{0s} = T_{s0} \ni - \frac{|q|}{M} \left( \frac{q_0-E_q}{M} \right)
$$</p>
<p>Naively, it appears to me that these cancel out the pole at $q_0 = E_q$, but the remaining portion does not vanish at at $q_0 = E_q$ (or at least as at $q_0$ approaches $E_q$) I think that a careful analysis of the behavior of these terms the pole might shed light on this, or that maybe it is a gauge artifact, but I am stuck.</p>
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<p>Is there formula that gives reflectance of very thin film of given metal (tens of nanometers) to the visible light of given wavelength(808nm) ? Which properties of metals are needed for the formula ?</p>
<p>I would like to draw a plot of reflectance that is a function of titanium film thickness. Thanks</p>
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<p>I'm looking at torque-free precession, occurring when the angular velocity isn't aligned with a principal axis of an object. I've looked at some Euler arguments, some decompositions of $\vec{\omega}$ into $\vec{\Omega} + \vec{\omega_0}$, where $\vec{\omega_0}$ is the rotation around the principal axis, etc.</p>
<p>I'm just looking in the inertial frame, for the moment.</p>
<p>I have a handle on why $\vec{\omega}$ must precess, but I'm trying to quantitatively describe the precession of a simpler system, two masses $m$ each on the ends of a massless rod of length $l$, with $\vec{\omega}$ neither parallel nor perpendicular to the rod.</p>
<p>No matter the orientation of $\vec{\omega}$, $\vec{L}$ is perpendicular to the rod (I think). I'm visualizing $\vec{L}$ as oriented along $\hat{k}$, with the rod therefore in the x-y plane and $\vec{\omega}$ tilted (not perpendicular to x-y).</p>
<p>I'm having trouble visualizing the motion of the rod as $\vec{\omega}$ precesses around $\vec{L}$. What would this look like? How do I calculate the inertia tensor for one of these arbitrary arrangements?</p>
<p>Thanks for the help!</p>
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<p>I have read that polarized light is treated by Jones vectors and that to treat partially polarized light you have to use Stokes vectors and mueller matrices.</p>
<p>Nonetheless, the optics notes that my professor have given us have no mention of mueller calculus, and we have assigned exercises involving partially polarized light passing through polarizers, retarders... so I figured that perhaps the following is legitimate:</p>
<p>The Stokes parameters characterizing partially polarized light are the following:</p>
<p>$s_1=Vs_0\cos{2\alpha}$</p>
<p>$s_2=Vs_0\sin{2\alpha}\cos{\delta} $</p>
<p>$s_3=Vs_0\sin{2\alpha}\sin{\delta} $</p>
<p>from the Stokes vector $(s_0,s_1,s_2,s_3)$ we get $\alpha$ and $\delta$ and build a Jones vector using:</p>
<p>$|e\rangle=\left(
\begin{array}{c}
\cos{\alpha}\\
\sin{\alpha}e^{-i\delta}
\end{array}
\right) $</p>
<p>and from here we go on using jones matrices.</p>
<p>Is this doable? And if it is, why do people use mueller matrices if this can be done?</p>
| 6,965 |
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] |
<p>I have always wondered about how <a href="http://en.wikipedia.org/wiki/Cosmological_constant" rel="nofollow">cosmological constant</a> is characterized. So since it is still a hypothesis you often read the “cosmological constant measured to be ….”. Shouldn't the statement read “cosmological constant calculated to be ….” . Or Is it that such semantics does not matter.</p>
| 6,966 |
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<p>In the literature, spin liquids are only possible in Mott insulators, however, I'm not entirely sure why the nuclear spin can't create a spin liquid in a band insulator.</p>
<p>Is this possible? If so, is there any specific reasons that it is not mentioned in the literature? If it is not possible, why not?</p>
| 6,967 |
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<p>To explain the expansion of space I have often heard people saying that space is continuously created. This picture is usually applied to cosmological scales but I`m nevertheless curious if some microscopic description of this process could be found (or probably is already available?).</p>
<p>The first thing which comes into my mind by thinking about this is that in theories like loop quantum gravity, there is an idea that space is quantized. Would space expansion then be explained as the creation of new quanta in loop quantum cosmology? Or are there other different ideas, such as the quanta themselves changing properties? Do people actually working on this already have some interesting results?</p>
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<p>I do not know how to calculate the direction, or unit vector of the force that appear between two magnetic field sources.
For example, let's assume I mean a current-carrying-wire by 'source of magnetic field'.</p>
<p><img src="http://t0.gstatic.com/images?q=tbn%3aANd9GcRdMJaUEnmPBVpQUAuhTDC-n_MDYjbgxzYqaBFPqBj0wUYMUCpN" alt="Image"></p>
<p>In the image above, I know there is no force on circular wire, but why?</p>
<p>I want to know if we can only calculate the direction using a function of current and magnetic field vectors $\vec{F}(\vec{i},\vec{B})$, or is there a way to calculate the direction as a function of magnetic field vectors those are generated by the wires, $\vec{F}(\vec{B}_1, \vec{B}_2)$?</p>
<p>Do we take into account of the unit vectors on the intersecting points of magnetic fields those are generated by the wires?</p>
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<p>In the Physics classes, the professor did an experiment using de Van de Graaff generator, by which he held a neon tube radially outward to the V d Graaff dome, and the neon lit up. I understood that this was because there was a potential difference between the 2 ends when placed radially which caused the electrons to flow, therefore lighting up the neon. My question is, since it was not a closed circuit, couldn't the electrons "run out"? I mean, that while under the electrical influence, there wasn't a way for them to return to the other end of the tube, and so, wouldn't eventually all of them be transferred to the other side?</p>
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<p>Stated: The atomic spectra of hydrogen and deuterium are similar however shifted in energies. </p>
<p>So im trying to explain why it is that the emission lines are shifted and how they are shifted.</p>
<p>Since the nucleus of deuterium contains both a proton and a neutron its notably heavier than the nuclueus of hydrogen, which only contains a proton. And since the transition energy is given by following equation:
$E_i-E_f = \frac{\mu_xZ^2e^4}{(4\pi\varepsilon_0)^22h^2}\left[\frac {1}{n_f^2}-\frac {1}{n_i^2}\right]$, where $\mu_x $ is the reduced mass of Atom $X$. its clear that the energy varies with the atom mass.</p>
<p>But i dont really know how to tackle the second part of my problem i.e. explaining how they are shifted. </p>
<p>Thank you in advance!</p>
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<p>I believe that we can take a single photon state as a tensor product of a frequency Hilbert space (infinite dimension) and a polarization Hilbert space (dim 2). Does this mean we can measure the polarization and the frequency at space-like separated regions a the same time?</p>
<p>My second question is about frequency and polarization entanglement. If we have an entangled state between the frequency and the polarization, but we model the system as having a classical distribution over frequencies, what kinds of mistakes will we make in understanding any joint measurements of polarization and frequency? I am thinking about a channel which entangles the polarization and the frequency which we imagine is a classical channel over frequencies.</p>
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<p>Assume we have two charged particles colliding.
He have particle 1 with mass $m_1$, charge $Z_1 \cdot e$ which travels in $x$-Direction passing by a STATIONARY particle 2 (mass $m_2$, charge $Z_2 \cdot e$) at distance $b$ (impact parameter).</p>
<p>I want to calculate the change in momentum in $y$-Direction (as the change in momentum in x-Direction vanishes). </p>
<p>$\Delta p_y = Z_1 e \int_{-\infty}^\infty F_y dt = Z_1 e \int_{-\infty}^\infty F_y \frac{1}{v}dx $ (as $dx/dt = v)$ in the script im trying to follow they now multiply by $1$ as follows:</p>
<p>$\Delta p_y = \frac{Z_1 e}{2 \pi b v} \int_{-\infty}^\infty 2 \pi b F_y dx$</p>
<p>They now immediately evaluate this theorem using Gauss-Theorem to: $\frac{Z_1 e}{2 \pi b v} \frac{Z_2 e}{\epsilon_0}$</p>
<p>Generally the Gauss-Theorem relates the integral over the surface to an integral over the space that surface encloses. I have a bit of a problem seeing over which surface/space area im integrating here, since its just an integral along the x-axis.</p>
<p>Thanks in advance for any help!</p>
<p>Cheers</p>
| 6,973 |
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] |
<p>Is it possible to see the oscillations with plasma frequency in a gas of particles of the same charge (not mixture of positive and negative charges)? </p>
| 6,974 |
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] |
<p>I am calculating the phase diagrams for Li-Mg binary alloy with reference to the following text:
<a href="http://ieeexplore.ieee.org/xpl/articleDetails.jsp?arnumber=5600701" rel="nofollow">http://ieeexplore.ieee.org/xpl/articleDetails.jsp?arnumber=5600701</a></p>
<p>In the equation used in the above paper, i am confused about whether G0 for a component is a function of temperature?</p>
<p><img src="http://i.stack.imgur.com/V78Oq.png" alt="enter image description here"></p>
| 6,975 |
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<p>What is difference between the miltary radar in 1940's from commercial antenna that is for the use of TV?</p>
<p>I have read article from some of the WW2 history website that call the German radar the antenna, so does the WW2 radar have features that is more powerful than today's antenna</p>
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<p>I am wondering if my line of thought is correct - and thus the resulting answer to the problem above would be correct.</p>
<p>As we know the gravitational force (of two point masses) is given by $$F = G\frac{m_1m_2}{r^2}.$$</p>
<p>So the gravitational force/vector field reduces with the distance squared. Now this is the formula in 3 spatial dimensions - and I always picture it as a point with gravitational field lines moving outward. Then the "strength" of the field would be the density of the lines. And hence the density drops with the distance squared (as it is inversely proportional to the area of the sphere at that distance).</p>
<p>Now taking this line of thought to other situations we can think of course about a hypothetical 2 dimensional world. Here gravity would also be. And here we can also see the density of the "gravitational field lines". However as they propagate only in 2 spatial dimensions the density would be inversely proportional to the circumference of the circle at a distance $r$. And hence the formula would lose the square and become like:</p>
<p>$$F = G\frac{m_1m_2}{r}$$</p>
<p>(With change $G$, and obviously we can't talk about mass in 2d).</p>
<p>Is this line of thought correct?</p>
| 91 |
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<p>I tryed to understand the Bohm interpretation and this is what picture appeared to me.
Please tell me if I understood something incorrectly.</p>
<ul>
<li><p>All particles have definite positions and follow deterministic rules of dynamics</p></li>
<li><p>Any future configuration of an isolated subsystem is only dependent on initial conditions</p></li>
<li><p>Even slight difference in initial conditions may result in huge differences in the result.</p></li>
</ul>
<p>The problem is that those initial conditions, are inherently unknown. This is fundamental: even if an observer manages to measure the whole Bohm state of the entire universe, he still would not know the Bohm state of himself.</p>
<p>This is like making predictions bout future states of a three-body system based on Newtonian mechanics with initial coordinates known only with finite precision. Due to apparent chaoticity of the solution the possible results may be dramatically different.</p>
<p>Correct me if I am wrong.</p>
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<p>The current definition of a second is stated <a href="http://en.m.wikipedia.org/wiki/Second" rel="nofollow">here</a> and I found a presentation on the <a href="http://www.bipm.org/" rel="nofollow">BIPM</a> site which discusses plans to change to a "better" definition of a second. You can find the presentation <a href="http://www.bipm.org/utils/common/pdf/RoySoc/Patrick_Gill.pdf" rel="nofollow">here</a>. The plan is to use a new definition based on "an optical transition". In what way does the current definition fall short? The BIPM presentation tries to explain why we need a new definition, but I don't have the background to understand it. </p>
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<p>How do I extend the general Lorentz transformation matrix (not just a boost along an axis, but in directions where the dx1/dt, dx2/dt, dx3/dt, components are all not zero. For eg. as on the Wikipedia page) to dimensions greater than 4?</p>
<p>Thanks</p>
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<p>I was wondering if for a point-like charged object, does the gradient of the electric potential point in the direction of maximum increase or maximum decrease of the function $V$? </p>
| 6,980 |
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<p>We know gravitational is continuously acting on us. But let us assume that we are hanging in the space alone away from anything to affect our position. And suddenly a giant planet appear's a few million km from us then we will feel it's gravitational force on our body. But I wanted to know will it work instantly or it will take some time to take effect just like light take some time to reach us ? </p>
| 4 |
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<p>I am manipulating an $nxn$ metric where $n$ is often $> 4$, depending on the model. The $00$ component is always tau*constant, as in the Minkowski metric, but the signs on all components might be + or - , depending on the model. (I am not trying to describe physics with this metric). Can I call this metric a Minkowski metric? Or what should I call it?</p>
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<p>If $i$ and $j$ are two variables then Kronecker delta is written as</p>
<p>$$\delta_{i,j}~:=~\begin{cases}1 \hspace{3mm} \mbox{if} \hspace{3mm} i=j,\\
0 \hspace{3mm}\mbox{if} \hspace{3mm}i \neq j. \end{cases}$$</p>
<p>same definition is used for orthogonality. </p>
<p>My question is when will we use Kronecker delta function and when will we use orthogonality?</p>
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<p>Another way of stating it: When you have leaves in a (cylindrical) tea cup, if you rotate the cup (eg clockwise), the leaves largely stay in place relative to the table, but spin clockwise a little. What factors go into determining how much force (spin? torque?) transfers to the tea leaves (or to a neutrally buoyant object inside the cup)</p>
<p>I'm looking for highly simplified mechanics that can be translated into real-time rigid-body 2D physics for a video game (eg circles containing neutrally-buoyant specimens of various shapes). Having the circles bounce around like billiard balls is easy enough but I have no idea how to make them appear to be specimen-filled.</p>
<p>Any help is appreciated as I have little clue what to search for. I'm trying to avoid complex/super-realistic models/simulations (eg one that would contain many particles), but I'd be interested in the physics of it all, all the same. </p>
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<p>I need to get the x/y values of a fired projectile given the angle of the initial power, and air resistance.</p>
<p>The formulas I have are</p>
<p>$$V_x=V_o \cos\theta$$</p>
<p>$$X=V_o \cos \theta t$$</p>
<p>$$V_y=V_o \sin \theta -gt$$</p>
<p>$$Y=V_o \sin \theta t-(1/2)gt^2$$</p>
<p>To calculate the initial velocity $V_o$, I did</p>
<p>$$V_o= \sqrt{2E/m}$$</p>
<p>where </p>
<p>$E$ = power with which the ball was fired,
$m$ = mass of ball</p>
<p>What I need is how to add air resistance to this equation</p>
<p>Besides, are my equations correct?</p>
| 6,984 |
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<p>I am trying to confirm the proof of Langreth's theorem / rules as seen in <a href="http://www.iue.tuwien.ac.at/phd/pourfath/node52.html" rel="nofollow">http://www.iue.tuwien.ac.at/phd/pourfath/node52.html</a> .</p>
<p>My problem is equation 3.55. I would do it like this:</p>
<p>$\int_{C_{1}} \mathrm{d}\tau A(t,\tau) B(\tau,t') = \int_{-\infty}^{t} \mathrm{d}t_1 A^{>}(t,t_1) B^{<}(t_1,t') + \int_{t}^{-\infty} \mathrm{d}t_1 A^{<}(t, t_1) B^<(t_1, t')$</p>
<p>And now I flip the integration boundaries on the right and replace $A^r = A^>-A^<$:</p>
<p>$ = \int_{-\infty}^{t} \mathrm{d}t_1 (A^{>}(t,t_1) - A^<(t,t_1) ) B^{<}(t_1,t') = \int_{-\infty}^{t} \mathrm{d}t_1 A^r(t,t_1) B^<(t_1,t)$</p>
<p>And thus my integration does not go from $-\infty$ to $\infty$ as shown in the source.</p>
<p>What am I missing?</p>
<hr>
<p>Alright, I got it. It works because of the definition of the retarded function with a theta function:
$$ A_r(t,t') = \theta(t-t') [ A^>(t,t') - A^<(t,t')] $$
So the retarded function is always zero when $t' > t$. One can therefore extend the integration to $\infty$.</p>
<p>I leave this here so it might help someone else later.</p>
| 6,985 |
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<p>Let's imagine a plastic container completely closed. We put alcohol inside it filling it at half. Now we make a hole to introduce 2 little Conductor one near the other so that they don't let the air pass. We make eletricity pass trough one of the conductor so that a Spark being created. </p>
<p>The question is: does the air+alcohol-gas burn creating a big pressure or does the container need of a little hole to burn?</p>
| 6,986 |
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<p>I am just looking on the TopQuark production via proton antiproton collision and strong interaction. There seem to be three basic possibilities. </p>
<ol>
<li><p>$q + \bar q \rightarrow Gluon \rightarrow t +\bar t $</p>
<p>This method is clear</p></li>
<li><p>$Gluon + Gluon \rightarrow Gluon \rightarrow t + \bar t$</p>
<p>Why do I need two Gluons to begin with? Why can't I just take one?</p></li>
<li><p>What is the particle going straight up/down in this image? <img src="http://i.stack.imgur.com/jM039.png" alt="enter image description here"></p></li>
</ol>
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<p>I have a paper which lies on a flat surface. The paper is fixed on one side and the opposite side can slide in the direction of the opposing side. As side end slides toward the other, a "bump" forms. I want to know what the solution is to the shape of this bump. There must be some standard solution for this case.</p>
<p><em>EDIT</em>: Luboš Motl did provide a really nice answer for the case when the paper is assumed to lie flat against the surface at the ends (also with the assumption that the bump is small).
I also found this <a href="http://sci-toys.com/bent_paper_problem.pdf">interesting paper</a> on a similar topic.</p>
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<p>I already understand that light cannot escape a black hole after passing the event horizon, so please do not explain that to me. What I would like to know is this:
a well known fact about light (a photon specifically) is that it travels at the speed of light, and at no other speed, which means that it has no rest mass, as an example, as it does not stop. As a photon aproached a black hole, it would begin to spiral around it as it got ever closer to the singularity at the centre. The closer the photon got to the singularity, the shorter the amount of time it would take to go once around the singularity, as it remained at its constant speed. However, on reaching the perfect centre, it would stop moving completely relative to the blackhole, so would no longer be travelling at the speed of light. Can you explain why this happens or (more likely) where I have gone wrong?</p>
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] |
<p>Related to this question
<a href="http://physics.stackexchange.com/questions/35/where-do-magnets-get-the-energy-to-repel">Where do magnets get the energy to repel?</a></p>
<p>If I have a magnet repelling another, eg one in my hand, the other being pushed along the desk, how do the each of the magnet's fields actually "push" against each other? What translates the magnetic field to kinetic energy?</p>
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<p>According to Bohr, electron revolves around the nucleus because of force of attraction between electron and proton. This force of attraction gives energy to the electron. So my question is this that- In which form does this electron get the energy?</p>
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<p>I have a spin system:
<img src="http://i.stack.imgur.com/iRS4g.png" alt="enter image description here"></p>
<p>As shown in the picture, there are two spins S1 and S2, and a pair of interactions between them. One is a ferromagnetic interaction and the other is anti ferromagnetic interaction. I am trying to calculate the Hamiltonian of this system. </p>
<p>The Hamiltonian of the system is:</p>
<p>$$ H = -J_F S1_z S2_z +J_{AF} S1_z S2_z $$</p>
<p>$S1_z$ is the spin matrix for Z direction for spin 1 and $S2_z$ is the spin matrix for Z direction for spin 2. If we allow two random values for $J_F$ and $J_{AF}$, -0.5 and 0.5 respectively the Hamiltonian of the system is as follows.</p>
<p>$$ H = 0.5 S1_z S2_z + 0.5 S1_z S2_z $$
$$ = S1_z S2_z $$
$$ =
\begin{pmatrix}
1 & 0 \\
0 & -1 \\
\end{pmatrix}
\times
\begin{pmatrix}
1 & 0 \\
0 & -1 \\
\end{pmatrix}
$$
$$ =
\begin{pmatrix}
1 & 0 \\
0 & 1 \\
\end{pmatrix}
$$</p>
<p>Am I able to calculate the Hamiltonian correctly?</p>
| 6,992 |
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<p>I'm working on a science fiction story that involves two spaceships engaged in combat while in orbit around a planetoid. My original idea called for spaceship A to trick spaceship B into firing a passive projectile such that it would orbit the planetoid and strike spaceship B. This was meant to happen within a matter of minutes, much faster than the spaceships' orbital period.</p>
<p>Of course, this was flawed because if the projectile travels with greater velocity it will simply have a larger orbit than the spaceships. Angling the trajectory closer to the planetoid would cause a slingshot effect, but that would be followed by a long, elliptical orbit, equally problematic.</p>
<p>It seems this scenario isn't possible without at least a third body. The story already calls for the battle to take place within a ring of asteroids (or rocky satellites, technically) also orbiting the planetoid. It occurs to me that the projectile could be made to slingshot around the planetoid, then slingshot around a large asteroid such that it quickly came back around and struck spaceship B.</p>
<p>The sheer improbability of this scenario doesn't concern me. Primarily, I just want to make sure such a trajectory would be <em>possible</em>. A secondary issue is that the planetoid is meant to be large enough to have small satellites orbiting it, but the flight of the projectile should be as quick as possible, only a few minutes in the original sketch. I'm worried that the "third body" asteroid can't be large enough to act as the second slingshot, but small enough to conceivably inhabit the ring of debris around the planetoid.</p>
<p>Is such a scenario possible?</p>
<hr>
<p>First time poster. If my question isn't specific or constructive enough, please let me know how it can be improved. Feel free to edit my question to improve terminology, etc.</p>
| 6,993 |
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<p>EDIT:Recently , I have been observing the moon and i realised the moon seems to be moving very quickly from one place to another, and back to its initial position in a short cycle(few minutes),the movement seems like it's moving back and forth about some axis at very small radius., I don't think this is due to orbital motion around Earth , conz the movement should be very insignificant given that a complete cycle is a calendar month. </p>
<p>Here is my theory: On the day I observed the moon was quite cloudy , I thought there were many clouds in front of the moon which caused uneven refraction due to uneven cloud layers with uneven thickness passing by, hence the moving line of sight slightly deviates . </p>
<p>However, I realise even I observe the moon directly in clear sky, it seems to move pretty quickly and back to its original position in a short time, why is it like that? Is it due a phenomena so called libration? </p>
<p>PS i'm observing the moon just by naked eye.</p>
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<p>Based on how light behaves when it passes through mediums, i.e. the wavelength of light changes when it passes through mediums of different refractive indexes, wouldn't it be possible to convert infrared light into visible light by passing it through a medium (e.g. $CO_2$)? Also, is it possible to make holograms from a device that works using this principle? </p>
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<p>This is an example problem with solution in a book I am reading on electrostatics. </p>
<p>"We have two square parallel plate capacitors with length $a$ and separation $d$ with $d<<a$. If we triple all dimensions of the capacitor, by what factor does the capacitance change?"</p>
<p>$C=\frac{\epsilon_0A}{d}$ and $A$ is the area. The book says the capacitance would change by a factor of $3$, but doesn't it change by a factor of $9$?</p>
<p>$C=\frac{\epsilon_0(a)^2}{d}\longrightarrow\frac{\epsilon_0(3a)^2}{d}= 9\frac{\epsilon_0(a^2)}{d}$</p>
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<p>With what we know about physics, is it possible that when the universe 'began', around when quarks and leptons were produced, another particle, which doesn't couple to either quarks, leptons or photons was also produced ? The only other way that we can observe its existence is via the effects of its gravitational field. In others words, some ''dark-matter-particle'' that doesn't interact with known forms of matter, except through gravity? </p>
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<p>I am having trouble understanding how to obtain a spacelike slicing of the Schwarchild black hole. I understand there is not a globally well defined timelike killing vector, so we can define t=cte slices outside the horizon and r=cte slices inside the horizon. </p>
<p>In the literature people define connector slices that join these two spacelike surfaces. </p>
<p>What is the formal definition of a spacelike connector slice? What is the most practical way to go about finding its mathematical expression?</p>
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<p>Let's say we have a normal circuit with a light bulb, with wires and a battery. </p>
<p>When one places a capacitor in this circuit, how is the light bulb able to light up, even when the capacitor prevents the flow of charge? Also, why does it dim and then go out eventually?</p>
<p>Then when the battery is removed from this circuit, how is the light bulb still able to light up? And what is happening when the light bulb dims and goes out in this situation as well?</p>
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<p>As we know, if an ideal gas expands in vacuum, as its energy is unchanged, the temperature remains the same. An ideal gas's energy does not depend on volume. In general, the energy is $kT$ times the total degrees of freedom, like in an ideal gas, the total degrees of freedom is $N$ particles plus three dimensions, $3N$. </p>
<p>Then if the total energy of the universe is $kT$ times the total degrees of freedoms of the universe, as the universe expands, its energy and entropy should not change, but if the temperature falls, the number of degrees of freedom should grow. It is quite puzzling to me that the universe is having more and more new degrees of freedom. It seems to be contradictory to the entropy argument.</p>
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<p>I'm going over some (molecular dynamics) related literature - specifically the derivation of the <a href="http://lib.jncasr.ac.in:6060/jspui/bitstream/123456789/570/1/464174.pdf" rel="nofollow">Weighted Histogram Analysis Method (WHAM)</a>. </p>
<p>As a quick backdrop WHAM is a method for stitching together conceptually overlapping and independent experiments done in Monte Carlo or Molecular Dynamic simulations. This isn't actually super important, as this is more a notation question. </p>
<p>Before the paper gets to the WHAM derivation, there's a discussion on the actual problem, where two variables are defined</p>
<p>$ x = \text{coordinates of atoms}$</p>
<p>$ \xi = \text{reaction coordinate - is a function of x} $</p>
<p>There's also a discussion on the notation used, where a caret above a value indicates it's a function,</p>
<p><strong><em>"Thus, $\hat{V_i}(x)$ denotes the function and $V_i$ a particular value the function takes"</em></strong></p>
<p>Having defined this and after some more work, the paper defines the probability density obtained from a simulation, which can be written in the following manner.</p>
<p>$P(\xi) = \langle\delta[\xi - \hat{\xi}(x)]\rangle$</p>
<p>For the record</p>
<p>$P(\xi) = e^{\big(-\beta W(\xi)\big)}$</p>
<p>Firstly, there's no explanation of what $\delta$ is <em>at all</em>, which makes me think it's probably some common operator. I don't think a Kronecker delta because that doesn't make sense in this context (and it lacks subscripts) but maybe?</p>
<p>Secondly, is there some common way to interpret a variable which can also be a function? When I initially read it I assumed that $\xi(x)$ was looking at deviation from the reaction coordinate the $x$ variable showed, but in hind-site I don't know why I thought that. </p>
<p>The reaction coordinate, traditionally, is some physical component of the system. It could be the distance between two residues, a specific orientation of all the atoms in the system, the distance of a specif residue to some arbitrary point etc. I'm not sure if that's the same here, or if it has some other, more specific meaning in this context. </p>
<p><strong>EDIT:</strong> I'm currently reading another paper in the literature (<a href="http://rickhouse.dyn.berkeley.edu/pubs/pdf/replica-exchange-wham.pdf" rel="nofollow">"Use of the Weighted Histogram Analysis Method for the "Analysis of Simulated and Parallel Tempering Simulations"</a> by Chodera et al.) which a colleague recommended and uses very different notation - hopefully I can use this to reconcile what's going on here...</p>
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<p>I'm struggling with a seemingly simple problem in 2D motion. Basically, the question is, given accelerations in $x$ and $y$ ($a_x$ and $a_y$) as well as the angular velocity ($\omega$), how can we find the trajectory of the motion? Also, how can we report the motion like a computer mouse, i.e. in the reference frame of the sensor?</p>
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<p>Once, I read that Einstein founded the special relativity theory by imagining how an observer moves at the speed of light.</p>
<p>How does this thought experiment work? How to reach from this imagination to the relativity of time?</p>
| 7,003 |
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<p>Whenever I light up a tube-light it makes 'ting' 'ting' sound every-time it blinks. </p>
<p>I am talking about this tube-light<br>
<img src="http://i.stack.imgur.com/j5j97.jpg" alt="enter image description here"> </p>
<p>Why is it so? </p>
<p>I think its because of sparking(inside glass tube)</p>
<p>similar sound in the 6th box hover and listen <a href="http://www.pond5.com/sound-effects/1/tube-lights.html#1" rel="nofollow">here</a></p>
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<p>I'm a master student working on networks analysis in general. A network is something that has nodes and there are links between the nodes. Nodes and links could have attributes. An evolving network is one that changes overtime (new nodes and links are added..etc). An example of that is Facebook. Nodes are users and links represent the friendship relationship. Users have attributes (gender, age ..etc). A Facebook network as you know is an example of a social network. </p>
<p>The issue is that so many people studied traditional evolving networks like social networks, the web, or transportation networks. Currently I'm looking for novel examples of evolving networks to study them. So I thought there might be some examples in <em>physics</em> that could represent some kind of an evolving network. </p>
<p>So my question: Can you give me examples in <em>physics</em> for evolving networks? I dunno, maybe collisions of particles of so ... etc.</p>
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] |
<p>If I hang a rope from two points that are at the same height above the ground, what is the mathematical function that describes the shape of the rope between the two points? Assuming the mass of the rope is evenly distributed between the two points.</p>
<p>Upon visual inspection, it appears to be parabolic, but looks can be deceiving. I want to be able to calculate how much rope I'll need to span the distance, assuming that I want it to sag by a given amount. If I know the function of the rope's shape, I think I can use an integral to calculate the length of the curve.</p>
| 7,006 |
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<blockquote>
<p><strong>Possible Duplicate:</strong><br>
<a href="http://physics.stackexchange.com/questions/14212/the-collision-of-phobos">the collision of Phobos</a> </p>
</blockquote>
<p>Mars has two moons: <a href="http://en.wikipedia.org/wiki/Phobos_%28moon%29" rel="nofollow">Phobos</a> and <a href="http://en.wikipedia.org/wiki/Deimos_%28moon%29" rel="nofollow">Deimos</a>. Both are irregular and are believed to have been captured from the nearby asteroid belt.</p>
<p>Phobos always shows the same face of Mars, because of the tidal forces exerted by the planet on its satellite. This same force causes Phobos to come increasingly closer to Mars, a situation that will cause their collision in about 50 to 100 million years.</p>
<p>How can I calculate, given appropriate data, the estimated time at which Phobos collides with Mars?</p>
| 242 |
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] |
<p>Can our moon qualify as a planet? With regard or without regard to the exact definition of the planet, can the moon be considered as planet as Mercury, Venus and Earth etc. not as the satellite of the planet Earth.</p>
| 7,007 |
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<p>I'm looking for a webpage or a book or reference that might give me a better (and hopefully more scientific) understanding of how the Bahtinov mask works (which I understand to be the same as diffraction around the mirror arms, but I don't fully grasp how that occurs). Can anyone think of any good links here?</p>
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<p>If it was possible for charge to assume arbitrary densities, like we often see electrostatic exercises, and one could spread charge density uniformly over a ring, then how one would, theoretically, distinguish between the situations when there is a current on the ring and when there is not? In both situations, the distributions of charges would be constant over the ring during time.</p>
| 7,009 |
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<p>Submitting a semi-conductor to stress leads to a deformation in the energy-bands, roughly described by:$$H_{ij} = {\cal{D}}_{ij}^{\alpha\beta}\;\epsilon_{\alpha\beta}$$
$\epsilon$ being the strain (linked to the stress by Hooke's law), $H$ the perturbation Hamiltonian to the Hamiltonian describing a stress-free semiconductor, $i,j$ being indexing the energy level of the previous "free" Hamiltonian.
Still, I lack intuition regarding the apparition of this term, it seems that compression enlarges the band-gap whereas dilation tightens it. Why do we have this behaviour? I have tried thinking on electrostatic arguments, the potential decreasing as $r^{-1}$ we do have an increase of energies in $r \rightarrow \alpha r$ for $\alpha < 1$ or also seeing the dilatation as a renormalization group transformation, basically going to a coarser grain (although there probably other pertinent length scales in an atomic lattice (spreading of the electronic orbitals, ...) which would make this argument wobbly). Cutting to the point, what is your hand-waved way of seeing it? Books on the subject of deformation potential don't really seem to offer intuition on it, more numerical values for specific materials.</p>
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<p>This is a question I asked in Maths SE, and it was suggested I ask it here. This is a direct copy of <a href="http://math.stackexchange.com/questions/466883/are-electrodynamic-problems-in-the-complex-plane-relevant-to-real-life">that question</a>.</p>
<p>I have been reading Tristan Needham's excellent Visual Complex Analysis. The end of the book deals almost entirely with physics, using symmetries of conformal mappings to generalise the famous method of images technique in electrodynamics. The method of images is used in finding the electric field due to a charge when a grounded surface (such as a sphere or plane) is nearby. (See e.g. <a href="http://en.wikipedia.org/wiki/Method_of_image_charges" rel="nofollow">Wikipedia</a>.)</p>
<p>However, the problems seem to have very little "real life" applications to me, the main problem being that the complex plane is two dimensional, whereas we live in a 3 dimensional world.</p>
<p>To see this problem concretely, the electrostatic force is goes like $F\sim \frac{1}{r^2}$ because the surface area of a ball of radius $r$ centred at the charge is proportional to $r^2$. However since the complex plane is 2 dimensional, a charge in the complex plane produces a field which goes like $\frac1r$. So any solution we find to a problem of this kind in the complex plane isn't relevant in 3d.</p>
<p>And this is my question, is there any physical application of this technique? Or is it completely irrelevant? </p>
| 7,011 |
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<p>My friend thinks it's because she has less air resistance but I'm not sure.</p>
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<p>I am in the process of conceptualizing an iOS or computer game (a modification of Tetris) that would be suitable for motivated high school students and university undergraduates. It is meant to be an educational game that will convey concepts. Any ideas of what quantum phenomena I could represent in the game that won't be too conceptually challenging or daunting (because the game needs to be entertaining as well as educational).</p>
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] |
<p>The spin of fundamental particles determines if they are bosons or fermions. The atoms also have bosonic or fermionic behavior, for example $\require{mhchem}\ce{^4He}$ has bosonic and $\ce{^3He}$ has fermionic statistics. Which quantity of atom determines its statistics? </p>
| 243 |
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<p>I'm trying to simulate an RLC circuit using transfer function. Circuit is there: <a href="http://i.stack.imgur.com/MC8ME.png" rel="nofollow">http://i.stack.imgur.com/MC8ME.png</a> (I'm a new user therefore I cannot post images)</p>
<p><img src="http://i.stack.imgur.com/W8N1q.png" alt="alt text"></p>
<p><em>But I can, L.Motl...</em></p>
<p>Main current (I) is the output and V is the input. So far, I'm stuck at this:</p>
<blockquote>
<p>TF = (Q1(s) + Q2(s)) / ((30 + s)(Q1(s) + Q2(s)) + 20 * Q1(s))</p>
</blockquote>
<p>I need to get rid of Q1 and Q2, but I cannot find a way.</p>
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] |
<p>All I found in Google was very broad. From a physics models perspective, why can photons emitted from a laser cut? Does this cut mean that the photons are acting like matter?</p>
| 7,015 |
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<p>Usually, it is said that systematic errors can not be handled in a well-defined way, unlike statistical errors. </p>
<p>My question(s): </p>
<p>A) How can systematical errors be calculated for any experimental device or experiment? Maybe does it involve a "calibration" of experimental devices?</p>
<p>B) I have heard that bayesian methods provide THE tool to estimate and calculate systematical errors. How is it done?</p>
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<p><a href="http://en.wikipedia.org/wiki/Energy" rel="nofollow">Energy</a> versus <a href="http://en.wikipedia.org/wiki/Free_energy" rel="nofollow">free energy</a> diagram. I haven't been able to find an adequate definition of these two terms in relation to each other. Could someone point me in the right direction, please?</p>
<p>From <a href="http://pubs.acs.org/doi/abs/10.1021/ed061p83" rel="nofollow">Borrell and Dixon, 1984 "Electrode potential diagrams and their use in the Hill-Bendall or Z-scheme for photosynthesis"</a>:</p>
<p><img src="http://i.stack.imgur.com/lzsNL.jpg" alt="enter image description here"></p>
<p>And their statement in the conclusions section:</p>
<blockquote>
<p>This article has outlined the use of electrode potential diagrams in a
simple system, and in the Hill-Bendall scheme for photosynthesis, for
which the diagram provides a basis for understanding the mechanism.
Emphasis has been placed (1) on the use of arrows to depict electron
transfer, with a rule to give the correct direction of the coupled
reactions; (2) <strong><em>on the fact that the diagram is a free-energy diagram
and not an energy diagram</em></strong>, and (3) on the fact that the energy
quantities corresponding to heights represent the maximum work
available or the minimum work required for particular processes.</p>
</blockquote>
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<blockquote>
<p>Sparked off by <a href="http://physics.stackexchange.com/questions/21813/is-sea-water-more-conductive-than-pure-water-because-electrical-current-is-tran/">Is sea water more conductive than pure water because "electrical current is transported by the ions in solution"?</a></p>
</blockquote>
<p>This question really belongs on chemistry.SE, which is still in area 51.</p>
<p>While answering this question, I realised that there was a flaw in the standard logic for these situations.</p>
<p>Let's take $NaNO_3$ solution in water, and compare it with pure water (same size cell). (I'm not taking $NaCl$ for a reason*). </p>
<p>In both cases, we have nearly the same(~7) PH and pOH, right? So concentrations of $H^+$ and $OH^-$ are the same, and thus contribution of these ions to the overall conductance/conductivity is the same. These ions both migrate to the electrodes, get reduces/oxidised, and emit/absorb electrons, facilitating passage of current.</p>
<p>OK. Now, let's consider the $Na^+$ and $NO_3^-$ ions. Yes, they migrate as well. But, they don't get redoxed (the water ions are preferentially redoxed). So, all I see happening here is a buildup of charges on either electrode, which will stop once equilibrium is attained. This buildup of charges cannot migrate to the outside circuit like a capacitor, as it cannot translate to electrons. But, these ions still make a significant contribution to the net conductance by Kohlrausch law.</p>
<p><strong>So how does a general good neutral electrolyte help water conduct electricity?</strong> I feel that it should conduct electricity at the same rate as water; but by Kohlrausch law, it clearly doesn't. And anyways, I've always heard that impure water conducts electricity better.</p>
<p><sup>*$NaCl$ has the issue of overvoltage of $O_2$, causing oxidation to of $Cl_2$, which complicates the situation. The $Cl^-$, being able to get oxidised, facilitate current and thus don't serve as a good example here.</sup></p>
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<p>Let's start from a very simple argument: If $A$ and $B$ are some operators, then I can write their product as<br>
$$AB = (A-\langle A\rangle)(B - \langle B \rangle) + \langle A \rangle B + A \langle B \rangle + \langle A \rangle \langle B \rangle$$
In a mean-field approximation, I would then go on to neglect the first product on the right hand side under the assumption that fluctuations around the mean are small and thus terms quadratic in fluctuations can be neglected.</p>
<p>So far, so good. But then I look at the typical electron-electron interaction term:
$$H_{int} = \frac{1}{2}\sum_{k,p,q,\alpha,\beta} V(q) c_{k+q,\alpha}^\dagger c_{p-q,\beta}^\dagger c_{p\beta} c_{k\alpha}$$
To form the operators $A$ and $B$, I have to pair a creation and destruction operator, but there are two different ways to do so. </p>
<p>For the sake of brevity, let's call the operators $c_1^\dagger c_2^\dagger c_3 c_4$. Then in the mean field approach one gets <em>four</em> different terms, one for each way to pick one creation and one destruction operator and everage over them:
$$c_1^\dagger c_2^\dagger c_3 c_4 \approx
-\langle c_1^\dagger c_3 \rangle c_2^\dagger c_4
-\langle c_2^\dagger c_4 \rangle c_1^\dagger c_3
+\langle c_1^\dagger c_4 \rangle c_2^\dagger c_3
+\langle c_2^\dagger c_3 \rangle c_1^\dagger c_4$$</p>
<p>Naively, I'd have expected a factor of 2 here, because here I used <em>two</em> ways of forming operators $A$ and $B$ (in the sense from above) to obtain my mean-field approximation.</p>
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<p>This maybe a very naive question. </p>
<p>I have just started studying CFT, and I am confused by why we have two separate parts of everything in CFT (operator algebras and hilbert space), the holomorphic and anti-holomorphic, which are decoupled from each other. We initially introduced $z$ and $\bar{z}$ as two independent variables instead of say $t$ and $x$ in two dimensions. But now, we have got two isomorphic parts, the holomorphic and antiholorphic (they may given by $z \to \bar{z}$ and $h \to \bar{h}$), then what extra info. does the anti-holomorphic parts provide? And how is all the information contained in just one independent coordinate $z$? Also a physical theory should be the tensor product of the verma modules? Why do we need both the parts, and what is the physical significance of each. </p>
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<p>When you have your X-Ray tube, and want to make X-Ray for your X-Ray scanner or whatever. Then you get electrons from the cathode, which hit the anode, and from that produce a spectrum of X-Rays due to Bremsstrahlung, and characteristic lines from electrons jumping from shell to shell.
This gives the continuous spectrum, and the sharp peaks from the characteristic lines.</p>
<p>My question is then: When you do an X-Ray, of a leg, or something, is it all the X-Rays that you use, both from bremsstrahlung and the characteristic lines, or do you somehow remove the continuous spectrum, or something, and only use the characteristic line X-Reay ?</p>
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<p>I have a question regarding an equation in Polchinski's "String Theory, Volume 1, An introduction to the bosonic string". The equation is (4.3.27) on p.135.</p>
<p>This section is about the brst-cohomology of the open bosonic string. My question in particular is about the N=1 level and the derivation of the cohomology.</p>
<p>By calculating $Q_B |\psi_1\rangle$ and eliminating all terms containing $c_0$ and $b_n$ where $n \geq 0$ since these annihilate the states, I end up with the following equation:</p>
<p>$$0 = Q_B |\psi_1\rangle = (c_{-1}L^{(m)}_{1} + c_{1}L^{(m)}_{-1})(e_\mu\alpha^\mu_{-1}+\beta b_{-1} + \gamma c_{-1})|0;\boldsymbol{k}\rangle$$</p>
<p>which by further multiplication simplifies to:</p>
<p>$$ 0 = (e_\mu c_{-1}L^{(m)}_{1}\alpha^\mu_{-1}+ \beta c_{-1}L^{(m)}_{1} b_{-1} + e_\mu c_{1}L^{(m)}_{-1} \alpha^\mu_{-1} + \beta c_{1}L^{(m)}_{-1} b_{-1} + \gamma c_{1}L^{(m)}_{-1} c_{-1})|0;\boldsymbol{k}\rangle $$</p>
<p>where $c_{-1}c_{-1}=0$ was used.</p>
<p>In Polchinski eq (4.3.27), this turns out to be:</p>
<p>$$ 0 = \sqrt{2\alpha'}(k^\mu e_\mu c_{-1} + \beta k_\mu \alpha_{-1}^\mu)|0;\boldsymbol{k}\rangle $$</p>
<p>I don't quite see how the Virasoro operators and the ghost operators act on the state to produce this result, other than that it is probably the first and fourth term that gives the result with the others being zero. I've been trying to figure it out, but I haven't been able to, would someone be able to enlighten me?</p>
<p>Thank you!</p>
| 7,022 |
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<p>I recently watch a lecture by Neil Tyson where he said the closest thing we have to a vacuum is interstellar space. I believe he said there will be one atom per 1 cubic meter or something close to that. I know that pressure equalizes, meaning that a higher pressure systems will move towards low pressure systems to reach an equilibrium. So I was wondering how that works. Do all the atoms in space try to get equidistant from one another to maintain reach an equilibrium pressure. If you made a 1 cubic foot sphere and put a single atom in it. Does pressure play a part there? Would the atom find the center of the sphere and stay there? Would it just bounce around unconcerned with equalizing pressure?</p>
<p>I'm sorry if this seems scatter brained. Whenever I watch a Neil Tyson video my brain is overloaded with curiosity.</p>
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] |
<p>Why does a spring lose a part of its energy when compressed for a long period of time? Is it because the material gets bent?</p>
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<p>why accelerated electrons "scatter" light?
From Yale's chemistry open class, the professor said when a light passes by an electron, it "pulls" the electron up and down, and emit radiation in all direction. How does this happen? And another question is about the X-ray diffraction. From the picture the textbook shows, when X-ray hits an atom, it's reflected, and because X rays hit lost of different layers , it's reflected separately. It's all about reflection, why we call that x-ray diffraction?</p>
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<p>I need to show that helicity is Lorentz invariant (under the proper Lorentz transformation) for the massless particles. I heard about most frequently used argument which contains an idea of impossibility to "outrun" the massless particle, so the sign of helicity is Lorentz invariant. But how about the absolute value of the helicity (when we don't divide this operator on the spin operator norm)? </p>
<p>I want to ask about the method of proof of Lorentz invariance of helicity value $h$, which is determined for the massless particles as
$$
W_{\mu} = hp_{\mu}, \qquad (1)
$$
or in the spinor language
$$
W_{c \dot {c}}\psi_{a_{1}...a_{n}\dot {b}_{1}...\dot {b}_{k}} = hp_{c \dot {c}}\psi_{a_{1}...a_{n}\dot {b}_{1}...\dot {b}_{k}}, \qquad (2)
$$
where $h = \frac{n - k}{2}$ and $\psi_{a_{1}...a_{n}\dot {b}_{1}...\dot {b}_{n}}$ has only one independent component.</p>
<p>Do expressions $(1), (2)$ give us the authomatical proof of the invariance of the helicity value? For example, the left part of $(2)$ transforms under spinor representation of the Lorentz group as the product of $n + 1$ undotted spinors and $k + 1$ dotted spinors, so the right side must transforms by the same way, so it means that $h$ is Lorentz scalar? Analogical thinking may be passed for $(1)$. </p>
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] |
<p>We embed the rotation group, $SO(3)$ into the Lorentz group, $O(1,3)$ : $SO(3) \hookrightarrow O(1,3)$ and then determine the six generators of Lorentz group: $J_x, J_y, J_z, K_x, K_y, K_z$ from the rotation and boost matrices.</p>
<p>From the number of the generators we realize that $O(1,3)$ is a six parameter matrix Lie group.</p>
<p>But are there any other way to know the number of parameters of the Lorentz group in the first place?</p>
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] |
<p>Who coined the word <a href="http://en.wikipedia.org/wiki/Permittivity" rel="nofollow">"permittivity"</a>? It appears that first usage was in 1887. Please cite your source. </p>
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0.023368287831544876,
0.034052636474370956,
-0.010636703111231327,
0.014241249300539494,
-0.027887115254998207,
0.032057736068964005,
-0.048896174877882004,
0.016550220549106598,
-0.004417014308273792,
0.010567606426775455,
-0.013184068724513054,
0.02587461657822132
] |
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