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<p>A particle of mass $m=0.5kg$, is attached to a spring of natural length $l=0.6m$ and modulus of elasticity $\lambda=60N$, and the setup is on a horizontal smooth table. The other end of the spring is attached to a fixed point $A$ on the table. The particle is then pulled so that the distance $AP=0.9m$, and is then released from rest. ($P$ is to the right of $A$, and the tension in the spring is $T$).</p>
<p>If you take $x$ (displacement from the centre of oscillation) as increasing to the right, then you can prove SHM if you can get equation that is like this:</p>
<p>$$a = -w^2$$
So
$$-T =ma$$ (The $T$ is negative because $x$ increases to the right making right positive, and $T$ is to the left, and $a$ is positive because in SHM $a$ is always in the direction of $x$ increasing.)
$$-\frac{\lambda x}{l} = ma$$
$$-\frac{60x}{0.6} = 0.5a$$
$$-200x = a$$ (which fits the equation at the top so its SHM.)</p>
<p>However if you take $x$ to increase to the left:</p>
<p>$$T = ma$$ ($T$ is positive because $x$ increases to the left making left positive and $T$ is to the left, $a$ is positive because, again, in SHM $a$ is always in the direction of $x$ increasing.)
$$\frac{\lambda x}{l} = ma$$
$$\frac{60x}{0.6} = 0.5a$$
$$200x = a$$ (which doesn't fit the equation because there is no negative.)</p>
<p>So I don't understand what I'm not getting right in the second part? how can changing the defining of the direction of $x$ increasing have such an effect?</p>
<p>(Instead of changing the direction of $x$ increasing I could have said the particle $P$ is pushed so that the spring is compressed and $AP=0.3m$ making the $T$ act towards the right which is the same direction as $x$ increasing).</p>
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] |
<p><a href="http://en.wikipedia.org/wiki/Maxwell%27s_equations" rel="nofollow">Maxwell</a>'s <a href="http://en.wikipedia.org/wiki/Mathematical_descriptions_of_the_electromagnetic_field" rel="nofollow">Equations</a> written with usual vector calculus are</p>
<p>$$\nabla \cdot E=\rho/\epsilon_0 \qquad \nabla \cdot B=0$$
$$\nabla\times E=-\dfrac{\partial B}{\partial t} \qquad\nabla\times B=\mu_0j+\dfrac{1}{c^2}\dfrac{\partial E}{\partial t}$$</p>
<p>now, if we are to translate into differential forms we notice something: from the first two equations, it seems that $E$ and $B$ should be $2$-forms. The reason is simple: we are taking divergence, and divergence of a vector field is equivalent to the exterior derivative of a $2$-form, so this is the first point.</p>
<p>The second two equations, though, suggests $E$ and $B$ should be $1$-forms, because we are taking curl. Thinking of integrals, the first two we integrate over surfaces, so the integrands should be $2$-forms and the second two we integrate over paths and so the integrands should be $1$-forms.</p>
<p>In that case, how do we represent $E$ and $B$ with differential forms, if in each equation they should be a different kind of form?</p>
| 8,061 |
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] |
<p>In the context of Cosmology models, What are examples of metrics with off-diagonal terms?</p>
| 8,062 |
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] |
<p>I've been having an unusually difficult time solving kinematics problems in comparison to all the other students in the classroom. It appears that I'm one of the unfortunate people with an anti-math mind (although I've been expecting as much for some time now). In short, I have no idea how to solve this problem:</p>
<blockquote>
<p>A carpenter tosses a shingle off a $9.4m$ high roof, giving it an initial horizontal velocity of $7.2 m/s$.<br>
a. What is the final vertical velocity of the shingle?<br>
b. How long does it take to reach the ground?<br>
c. How far does it move horizontally in this time? </p>
</blockquote>
<p>I tried to organize the data I had and the data I needed to calculate:</p>
<p><strong>Horizontal:</strong><br>
1. time is unknown and the same as vertical time<br>
2. initial position is $0m$<br>
3. final position is unknown<br>
4. initial velocity is $7.2m/s$<br>
5. final velocity is $7.2m/s$<br>
6. acceleration must be $0$ because velocity doesn't change</p>
<p><strong>Vertical:</strong><br>
1. time is unknown and the same as vertical time<br>
2. initial position is $12m$<br>
3. final position is $0m$<br>
4. initial velocity is gravity, or $9.81m/s$<br>
5. final velocity is unknown<br>
6. acceleration is $9.81m/s^2$</p>
<p>However, according to my unknowns and my equations, I cannot ultimately solve any of these because I cannot solve for time, which requires knowing distance, velocity, and acceleration, like this:<br>
$t = -v_i \pm \sqrt{v_i^2 + 2a(\pm d)} \over{a}$ Is this correct and is there another way to find time with less unknowns?</p>
<p>And none of my equations can explicitly find the initial or final positions of anything:</p>
<blockquote>
<p>$\Delta x = v_{ix} \Delta t + 0.5a_x \Delta t^2$ (also works with $y$)<br>
$v_{fx}^2 - v_{ix}^2 = 2a_x \Delta x$ (also works with $y$)<br>
$v_{fx} = v_{ix} + a_x \Delta t$ (also works with $y$)<br>
$\Delta y = 0.5(v_{fy} + v_{iy}) \Delta t$<br>
$y = v_{avg} * t$<br>
$\Delta y = 0.5(V_{fy} + V_{iy}) \Delta t$</p>
</blockquote>
<p>What I want to know is <strong>how to correctly solve this (with equations listed)</strong>. Please do not just provide the answer; thanks in advance.</p>
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<p>As in the title, I'm wondering is it possible.</p>
<p>I think it is possible, because we have powerful enough radiotechniques and gamma radiation are just EM waves, not particles. However I think is useless, because it costs too much.</p>
<p>Can anyone say something more about this? </p>
<ul>
<li>How much power would consume such emitter?</li>
<li>Would it be helpful anywhere?</li>
<li>How huge frequencies of current would it use in the antenna?</li>
<li>How big device would it be?</li>
</ul>
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<p>The entropy of an isolated system always increases,</p>
<p>Considering an intelligent actor in the system who can organise different objects in the system, doesnt the measure of disorder reduce albeit for short periods?</p>
| 265 |
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] |
<p>Does the speed of light apply to the speed of (waves?) in every known field, or does it only apply to the electromagnetic field?</p>
| 8,065 |
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<p>Since quantum Gauge theory is a quantum mechanical theory, whether someone could explain how to construct and write down <strong>the Hilbert Space of quantum Gauge theory with spin-S</strong>. (Are there something more rich/subtle than just saying the Hilbert Space are composed by the state space of infinite many sets and infinite many modes of harmonic oscillators? - i.e. more rich/subtle than usual spin-0 scalar fields' Hilbert space?)
Is the quantum Gauge theory's Hilbert Space written <strong>in a tensor product form or not</strong> (eg. thinking about put this gauge theory <strong>on the lattice</strong>)?</p>
<p>Whether there are differences for this procedure construction of Hilbert space for these three cases: </p>
<p>(1) spin-1 quantum Gauge theory with Abelian $U(1)$ symmetry</p>
<p>(2) spin-1 quantum Gauge theory with non-Abelian (such as $SU(N)$) symmetry</p>
<p>(3) spin-2 quantum Gauge theory (Gravity? or anything else)</p>
<p>Also, whether gauge-redundancy plays any roles? Is there similar thing like <a href="http://en.wikipedia.org/wiki/Faddeev-Popov_ghosts" rel="nofollow">Faddeev-Popov ghosts</a> happened in the path integral formalism, when one dealing with gauge-redundancy? </p>
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<p>I subconsciously feel not fully comfortable about <strong>Wick rotating or analytic continuation from Euclidean to Minkowski space</strong>. I simply wonder whether there is any subtlety here, and when we need to be conscious whether <strong>(i) this continuation can be dangerous or the continuation cannot be done; or (ii) the two theories(Euclidean and Minkowski) may not be the same?</strong></p>
<p>For example, if you read <a href="http://rads.stackoverflow.com/amzn/click/0521633036">Polchinski's String theory</a>, most of worldsheet and CFT business is done in Euclidean signature. But in Chap 3 of Polchinsk's at p.83, says ``The same procedure works for the Polyakov action if we write the metric in terms of a tetrad, and make the same rotation. <strong>This provides a formal justification for the equivalence of the Minkowski
and Euclidean path integrals.</strong> It has been shown by explicit calculation that they define the same amplitudes, respectively in the light-cone and conformal gauges.'' </p>
<p>But right at p.83 footnote, says ''<strong>In more than two dimensions, things are not so simple because the Hilbert action behaves in a more complicated way under the rotation. No simple rotation damps the path integral. In particular, the meaning of the Euclidean path integral for four-dimensional gravity is very uncertain.</strong>''</p>
<p>Is Polchinski providing an example here? <strong>Can someone explain in what (generic) scenarios there will be subtlety about the analytic continuation?</strong></p>
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<p>I am a bit lost how one has deduced the formula for electric field with electric dipole because of some inconsistency between different sources. The <a href="http://en.wikipedia.org/wiki/Dipole#Field_from_an_electric_dipole" rel="nofollow">Wikipedia article</a> contains a delta function in the formula, but <a href="http://phys.columbia.edu/~nicolis/Dipole_electric_field.pdf" rel="nofollow">this paper</a> gives only the formula without the delta function,</p>
<p>$$\bar{E} = -\nabla \phi = \frac{-p}{4\pi \epsilon_{0} r^{3}} ( \hat{e}_{z}-3\hat{u}_{r}(\hat{u}_{r} \cdot \hat{u}_{z}) ) $$</p>
<p>Is it intentionally left out?</p>
<p>My notes mention that the dipole is with a circular thing $P=Q d$ where $d$ is distance and $Q$ is charge.</p>
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<p>I was reading up on <a href="http://en.wikipedia.org/wiki/De_Sitter_space" rel="nofollow">De Sitter space</a>s, which states that the gravitational effects from a black hole is indistinguishable from any other spherically symmetric mass distribution. This makes a lot of sense to me.</p>
<p>I'm now super curious, can we just formulate all the properties of a black hole beyond the limit at which GR is needed in a completely Maxwellian / Newtonian sense? The Wikipedia article on the <a href="http://en.wikipedia.org/wiki/Kerr-Newman_black_hole" rel="nofollow">Kerr-Newman metric</a> seems to indicate so, but the equations are in terms of the GR metrics. This is not what I want, I want a simplification, a limit-case of that math.</p>
<p>Ted Bunn answered a part of my question in <a href="http://physics.stackexchange.com/questions/12169/detection-of-the-electric-charge-of-a-black-hole">Detection of the Electric Charge of a Black Hole</a>. Let me repeat the stupidly simple form of the gravitational and electric field for a black hole beyond the point at which GR is needed.</p>
<p>$$\vec{g} = \frac{GM}{r^2} \hat{r}$$</p>
<p>$$\vec{E} = \frac{Q}{4\pi\epsilon_0 r^2} \hat{r}$$</p>
<p>Correct me if I'm wrong, but these would be a meaningful and accurate approximation in many, in fact, most situations where we would plausibly interact with a black hole (if we were close enough that these are no longer representative, we'd be risking a date with eternity).</p>
<p><strong>Question:</strong> Fill in the blank; what would the magnetic field $\vec{B}$ around a black hole be?</p>
<p>Here is why I find it non-trivial: Every magnet in "our" world has some significant waist to it. So here is a normal magnet.</p>
<p><img src="http://i.stack.imgur.com/l7sr2.png" alt="normal magnet"></p>
<p>What happens when this is a black hole? Would the approximation for $\vec{B}$ that I'm asking for have all the magnetic field lines pass through the singularity? Or would they all pass through the event horizon radius but not necessarily a single point?</p>
<hr>
<p>I think most people who have understood this already, but ideally the answer would use the 3 fundamental metrics of a black hole. Mass $M$, charge $Q$, and angular momentum $L$. The prior equations for gravity and electric field already fit this criteria. So the answer I'm looking for should be doable in the following form.</p>
<p>$$\vec{B} = f \left( M, Q, L, \vec{r} \right) $$</p>
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] |
<p>How to derive the selection rules $\Delta L= \pm 1$, $\Delta S=0$ for electron spectroscopy?</p>
| 8,070 |
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<p>I will start with an example. Consider a symmetry breaking pattern like $SU(4)\rightarrow Sp(4)$. We know that in $SU(4)$ there is the Standard Model (SM) symmetry $SU(2)_L\times U(1)_Y$ but depending on which vacuum we use to break this symmetry, in a case you can totally break the SM symmetry, with the vacuum :
$$\Sigma_1 = \begin{pmatrix}
0& I_2 \\
-I_2 & 0
\end{pmatrix}$$
and in another case, these symmetry is preserved, with the vacuum
$$\Sigma_2 = \begin{pmatrix}
i\sigma_2& 0 \\
0 & i\sigma_2
\end{pmatrix}$$
In the first case (with $\Sigma_1$), the generators corresponding to the SM symmetry are part of the broken generators so the SM symmetry is totally broken.
In the second case ($\Sigma_2$), the SM generators are part of the unbroken generators then the SM symmetry is preserved. As you can read, I know the answers but not how to find them !</p>
<p>So, my questions are : </p>
<ol>
<li><p>How is it possible in general (not only for the $SU(4)\rightarrow Sp(4)$ breaking pattern) to construct the vacuum that breaks the symmetry ?</p></li>
<li><p>Is it possible, when constructing the vacuum, to ensure that the vacuum will (or not) break a sub-symmetry like the SM symmetry in the previous example ? </p></li>
</ol>
| 8,071 |
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<p>This is a very simple question, but I can't find an answer anywhere.</p>
<p>Earth's gravity at sea level is 1 (approximately 9.8m/s^2)</p>
<p>What happens being under sea level in a cavern/chasm/ravine, how much does gravity increase normally here on Earth?</p>
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<p>I ran into this while looking at measures of humongous amounts of data. How does the information (data) collected in an inverse femtobarn exposure compare to a gigabyte of data ?</p>
| 8,073 |
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<p>I'm having some trouble identifying a reference Landau and Lifshitz make in volume 3 - non-relativistic quantum mechanics - of their Course of theoretical physics. This is the page from the Oxford, Pergamon, 1977 edition:</p>
<p><img src="http://i.stack.imgur.com/OqC2L.png" alt="enter image description here"></p>
<p>There isn't a bibliography either at the end of the chapter or the book, and other references are usually well identified in footnotes. I know it's a long shot to identify and obtain a copy of this article (in English), but I would definitely like to have a look at it. It appears to be the original publication of Landau's method of complex-contour integration to find WKB-approximation matrix elements.</p>
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<p>Explanations I have read of why photons are emitted from atoms mention electrons being 'excited' to another energy level, and then returning to their base level, releasing a photon. I have also seen the occasional mention of 'fields' and I vaguely expect from what I've read that interaction of the electrons with these fields has some effect. I've also seen, but not used, Feynmann diagrams that involve additional particles.</p>
<p>Please can you (ideally) clarify what are the different theories and experimental evidence relating to electrons and photons because of changes to electron behavior?</p>
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<p>I'm looking for an explanation on why darker light filters let through less light than lighter filters (secondary/high school level). Although this seems to the layman like the obvious outcome, our science teacher was unable to explain why a darker blue and darker green filter let through less light than a yellow or turquoise filter.</p>
<p>Thanks</p>
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<p>This question is more practical than theoretical, but I am interested in the theoretical considerations as well.</p>
<p>My wife just bought a Samsung S3 phone with a 8 MP image sensor hiding behind a tiny lens. In daylight the pictures come out fine, but it suffers horribly in low-light conditions. Is there a theoretical limit as to how fine an image sensor can be behind a lens of a specific aperture, given a reasonable amount of ambient light and a reasonable shutter speed? Will increasing the sensor resolution beyond this limit decrease the actual resolution (the ability to resolve two points as individual points) of the final image?</p>
<p>Thanks.</p>
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<p>According to Poincare's Lemma, if $U\subset \mathbb{R}^n$ is a <a href="http://en.wikipedia.org/wiki/Star_domain" rel="nofollow">star-shaped set</a> and if $\omega$ is a $k$-form defined in $U$ that is closed, then $\omega$ is exact, meaning that there's some $(k-1)$-form, say $\eta$ with $\omega = d\eta$. Now, translating to vector fields, if we consider $U$ a star-shaped set in $\mathbb{R}^3$ and if $B$ is a vector field inside $U$ such that $\left\langle \nabla, B \right \rangle= 0$, then there's some vector field $A$ defined in $U$ such that $B = \nabla \times A$.</p>
<p>I've heard that <a href="http://en.wikipedia.org/wiki/Closed_and_exact_differential_forms#Poincar.C3.A9_lemma" rel="nofollow">Poincare's Lemma</a> turns out to be true even if $U$ is not star-shaped, but rather, simply connected. Now, in the hypothesis of Poincare's Lemma, the fact that the magnetic field satisfies $\left\langle \nabla, B\right\rangle = 0$ implies the existance of the vector potential $A$, with $B = \nabla \times A$. But now, what happens if the magnetic field is defined in some region of space that is not simply connected? In that case, the vector potential could not exist according to Poincare's Lemma (it doesn't says that it doesn't exists, but it doesn't guarantees the existance).</p>
<p>So, if the region where the field is defined has holes in it, what happens? Is really a chance that the vector potential doesn't exists? In that case, what are the physical consequences of that? Since I was always told that the vector potential was just a mathematical tool introduced to make life easier, I think that there wouldn't be so great impact on the point of view of conceptual explanation of the situation, however, I'm not sure.</p>
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<blockquote>
<p>A bar magnet is released from rest along the axis of a very long,
vertical copper tube. After some the the magnet :</p>
<p>(a) will stop in the tube.<br>
(b) will move with almost constant speed.<br>
(c) will move with an acceleration g.<br>
(d) will oscillate.</p>
</blockquote>
<h2>Attempt:</h2>
<p>Let the magnet be dropped with its north pole downward. We can consider long tube to be made of many circular loops joined together. At an instant $t=0$, the force mg acts on the magnet. $m$ is its mass. As it moves, by Lenz's law, each circular loop in the tube opposes its downward motion and hence upward force will act on it by many such loops. A time will come when the magnet experiences zero net force. Even eddy currents are produced in the magnet. The net force is $F-mg$ where at some instant will be nil, i.e. $F=mg$ as I already stated. Even if it has constant velocity at this instant magnetic flux by it linked with those coils is still varied and hence upward force still acts. Hence after this $F-mg<0$, and magnet accelerated upwards. Now again by Lenz' law coils oppose upward motion of magnet and this process continues till then again it move down.. and so on it just keep on oscillating...</p>
<p>So answer should be (d). But its wrong !! How ? Correct answer given in (b). Can anyone point out the flaw in my attempt ? </p>
| 8,079 |
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<p>I know that in free space, the width of a Gaussian beam can be written as $W=W_0\sqrt{1+(\frac{z}{z_0})^{2}}$. However, I was wondering if it was possible to express this width as a function of refractive index instead (since I don't believe a Gaussian beam originating in say, glass, will diverge in the same manner as one in air). Anyone have any ideas?</p>
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<p>From the perspective inside a black hole: Is information about everything outside a black hole - the rest of the cosmos - represented on the inside of the (event) horizon too?
NB. I realize it is the notion of a so called stretched horizon that is being used about the 2d-like sheet that information is scrambled on, but I haven't heard about its view-from-the-inside-analogy, that naively being a horizon hovering just "below" the event horizon; therefore the simplified use of "event horizon" in the question. I also realize that it would be impossible to sample (all) this information as an inside job, which I would suppose could have some bearing on the answer to the question. </p>
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<p><a href="http://en.wikipedia.org/wiki/Gamma-ray_burst" rel="nofollow">Gamma-ray bursts</a>(GRBs) are flashes of gamma rays associated with extremely energetic explosions</p>
<p>why nasa interested to Gamma ray burst?</p>
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<p>I've been thinking about the following problem for some time now and was wondering if anyone could shed any light on it.</p>
<blockquote>
<p>At time $t = 0$, turtle <code>A</code> sits at $(0,0)$ and turtle <code>B</code> sits at $(d,0)$. <code>B</code> begins moving straight upwards perpendicular to the $x$-axis with some constant velocity $v$. Simultaneously, the <code>A</code> turtle begins moving with the same constant velocity in such a way that its nose is always pointed at <code>B</code>. What is the distance between these turtles as $t \rightarrow \infty$?</p>
</blockquote>
<hr>
<p><strong>Edit</strong></p>
<p>So basically I've arrived at a nasty system of differential equations. Here's my reasoning:</p>
<p>We know that turtle <code>A</code> will always point towards turtle <code>B</code>, so if $(x(t),y(t))$ is the curve traced out by <code>A</code>, then we can determine $dy/dx = \dot{y}/\dot{x}$ at each point:</p>
<p>$\dot{y}/\dot{x} = (vt - y)/(d - x).$</p>
<p>The fact that <code>A</code> has a velocity $v$ gives rise to a second equation:</p>
<p>$\dot{x}^2 + \dot{y}^2 = v^2.$</p>
<p>If we solve these, the problem would be reduced to taking a limit.</p>
<p>Anyway, this is what I have so far - not much really. Would this be a good/feasible approach to pursue so as to solve the problem? Or is there a better way?</p>
| 8,083 |
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<p>I have heard that the traditional resistor-based fan speed regulators are inefficient. In fact, I have noticed that such regulators tend to get hot when the fan is set to low speed. However, does that mean that they save <strong><em>no power at all</em></strong> at low speeds, or is it just that power is saved but the amount of power saved is far less than ideal? I was expecting that since power P = V / R^2, the overall power consumption of the fan plus regulator system would be comparatively lower at lower speeds.</p>
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<p>The top rated answer to <a href="http://physics.stackexchange.com/questions/8850/alcubierre-drive-clarification-on-relativistic-effects">this question about the Alcubierre drive</a> asserts, "spacetime can dynamically evolve in a way which apparently violates special relativity," but according to <a href="http://en.wikipedia.org/wiki/Speed_of_gravity" rel="nofollow">the Wikipedia article on the speed of gravity</a>, changes in the gravitational field propagate at the speed of light. The second answer under <a href="http://physics.stackexchange.com/questions/5456/the-speed-of-gravity">this question</a> puts the limitation in broader terms: "distortions of spacetime are as limited to travel to the speed of light as any other physical influence."</p>
<p>These statements appear to contradict one another. My question is:</p>
<p>Does the Alcubierre drive depend on space-time distortions propagating at superluminal speeds, and is that possible under general relativity? If Alcubierre drive distortions of space-time can propagate at superluminal speeds under GR then why can't gravitational waves (another traveling distortion of space-time) also propagate at superluminal speeds?</p>
| 8,085 |
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] |
<blockquote>
<p><strong>Possible Duplicate:</strong><br>
<a href="http://physics.stackexchange.com/questions/2066/why-does-adding-solutes-to-pure-water-lower-the-the-specific-heat">Why does adding solutes to pure water lower the the specific heat?</a> </p>
</blockquote>
<p>Why do higher concentration salt solutions heat up more quickly?</p>
| 266 |
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<p>I was flicking through Zettili's book on quantum mechanics and came across a 'derivation' of the momentum operator in the position representation on page 126. The author derived that $\langle\vec{r}|\hat{\vec{P}}|\psi\rangle = -i\hbar\vec{\nabla}\langle\vec{r}|\psi\rangle$ (I've omitted the full derivation). However, from this relationship he concluded that $\hat{\vec{P}} = -i\hbar\vec{\nabla}$. I'm sure this is very basic but why can you immediately conclude this? Surely this assumes that $\langle\vec{r}|(-i\hbar\vec{\nabla})|\psi\rangle = -i\hbar\vec{\nabla}\langle\vec{r}|\psi\rangle$. I'm not sure why this is necessarily true. </p>
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<p>I'm an undergraduate mathematics student trying to understand some quantum mechanics, but I'm having a hard time understanding what is the status of the Schrödinger equation.</p>
<p>In some places I've read that it's just a postulate. At least, that's how I interpret e.g. the following quote:</p>
<blockquote>
<p>Where did we get that (equation) from? Nowhere. It is not possible to derive it from anything you know. It came out of the mind of Schrödinger. -- Richard Feynman</p>
</blockquote>
<p>(from the Wikipedia entry on the Schrödinger equation)</p>
<p>However, some places seem to <em>derive</em> the Schrödinger equation: just search for "derivation of Schrödinger equation" in google.</p>
<p>This motivates the question in the title: Is the Schrödinger equation derived or postulated? If it is derived, then just how is it derived, and from what principles? If it is postulated, then it surely came out of somewhere. Something like "in these special cases it can be derived, and then we postulate it works in general". Or maybe not?</p>
<p>Thanks in advance, and please bear with my physical ignorance.</p>
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<p>The hypothesis of confirmational holism ( <a href="http://en.wikipedia.org/wiki/Confirmation_holism" rel="nofollow">http://en.wikipedia.org/wiki/Confirmation_holism</a> ) speaks to all sciences, but seems especially trenchant in fundamental physics, the would-be end of the road for any reductionist (or modified reductionist) account of the world. Ironically, but naturally, the most fundamental physics comes at the top of any typical pedagogical climb -- knowledge at the smallest scales is the hardest won and most remote from ordinary experience. That makes typical textbooks useless for the philsophically-minded reader -- they all start with a raft of scientific assumptions that no sane physics major or graduate student would dare to challenge. I don't expect a complete account of the bootstrap from plain sense observation to fundamental laws because no one thinks as of yet that we know those fundamental laws. Nevertheless I seek to read a careful and skeptical analysis of the present situation. </p>
<p>PS I'd like to add some tags -- at the least philosophy-of-physic, confirmation-holism -- but lack the necessary reputation.</p>
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<p>When solving the hydrogen atom, as a 2 body problem, we have the motion of the center of mass and the motion relative to the center of mass. The well known energy spectrum, $E_n$, that goes like $1/n^2$ is the one resulting from studying the motion relative to the center of mass.</p>
<p>Now the energy of the center of mass of the atom, which is purely kinetic, should be added to $E_n$ to get the full energy. So unless the atom is at rest, there will be an extra (constant) term to $E_n$</p>
<p>My questions are, in practice when people study the hydrogen spectrum experimentally:</p>
<p>1-do people look at the spectrum from 1 atom only? if this is the case, then how can they guarantee that it is not moving (to properly identify the $1/n^2$ behavior)? (very cold atom technology was not known back in the day!)</p>
<p>2-if they are studying a gas of hydrogen, how come the spectrum to be detected is not washed out by the transitions coming from so many atoms in all directions?</p>
<p>3-again for looking at hydrogen gas, how the spectrum can be studied even to see if it goes like $1/n^2$ if the atoms are moving in all directions like crazy?</p>
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<p>Most of the Bodies and Objects in space are <em>likely to have</em> an <strong><a href="http://en.wikipedia.org/wiki/Atmosphere" rel="nofollow">atmosphere</a></strong>. Since the space is empty, the gases in atmosphere should have either <em>dissolved or emptied</em> into space. But, Why doesn't this happen? Still those objects are revolving with their atmosphere...</p>
<ul>
<li>Taking into account of Gas Giants - Since their atmospheres are much <strong>denser</strong>, there should be some pressure difference (like more pressure at highly denser areas and vice-versa). There are possibilities for the molecules to escape at highly denser regions of the atmosphere. </li>
</ul>
<p>Well, Is this RIGHT...?
Also, Answers based on <a href="http://en.wikipedia.org/wiki/Atmospheric_escape" rel="nofollow">Atmospheric</a> & <a href="http://en.wikipedia.org/wiki/Hydrodynamic_escape" rel="nofollow">Hydrodynamic</a> escape would be appreciated...</p>
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<p>E.g. if I know that my topology is that of a hyperboloid, how much freedom do I have left for my choice of the metric?
And the other way around: if my metric is some conformal factor times the unit hyperboloid how much freedom do I have left in my choice for the topology? Am I forced to conclude that the topology of this (Lorentzian) spacetime needs to be that of a hyperboloid?</p>
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<p>The Ising model is a mathematical model of ferro-magnetism in statistical mechanics. The model consists of discrete variables that represent magnetic dipole moments of atomic spins that can be in one of two states (+1 or −1). The spins are arranged in a graph, usually, a lattice, allowing each spin to interact with its neighbours. The model allows the identification of phase transitions, as a simplified model of reality <a href="http://en.wikipedia.org/wiki/Ising_model" rel="nofollow">[Source]</a></p>
<p>The algorithm I am using to calculate it is:
$$ E = -\sum\limits_{\langle i,j \rangle}^{n}J_{ij} s_i s_j - \mu H\sum\limits_{i}^{n} s_i $$</p>
<p>Where $E$ is energy, $J$ is the exchange constant, $s_i$ & $s_j$ represent nearest neighbour pairs (Like $s_{<1,0>} s_{<2,0>}$ or $ s_{<5,4>} s_{<5,5>} $). $H$ is the external magnetic field and $\mu$ is the magnetic moment. Usually H is given in units of $\frac{H}{\mu}$ and J is assumed to be isotropic, so the equation simplifies to:</p>
<p>$$ E = -J\sum\limits_{\langle i,j \rangle}^{n} s_i s_j - m\sum\limits_{i}^{n} s_i $$ where $m= \frac{H}{\mu}$.</p>
<p>This model can be used to predict the temperature at which a substance looses its ferromagnetism, which for cobalt is 1388K, for iron its 1043K and for nickel its 627K. This change from ferromagnetic to paramagnetic is found by observing a second order phase transition. (A continuous, dramatic change in the slope of the magnetism).</p>
<p>If you use $J=1$, $T_C \approxeq 2.27K$. How could I determine the exchange constants for other substances, if I know their internal molecular lattice structure, or must I determine it numerically, knowing $T_C$?</p>
| 8,092 |
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<p>If I understand this right the Ricci flow on a compact manifold given by </p>
<p>$\partial g_{\mu \nu} = - 2R_{\mu \nu} + \frac{2}{n}\!R_{\alpha}^{\alpha} \,g_{\mu \nu}$</p>
<p>tends to expand negatively curved regions and to shrink positively curved regions.</p>
<p>Looking at the above definition I`m wondering if the parameter n can be used to achieve $\partial g_{\mu \nu} = 0 $ even if the Ricci tensor is not zero such that the validity of physics, that depends on the metric to be constant (as a precondition), could be extrapolated to curved manifolds to describe an expanding universe with a positive cosmological constant?</p>
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] |
<p>How do you effectively study physics? How does one read a physics book instead or just staring at it for hours? </p>
<p>(Apologies in advance if the question is ill-posed or too subjective in its current form to meet the requirements of the FAQ; I'd certainly appreciate any suggestions for its modification if need be.)</p>
| 115 |
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<p>This year, the racetrack at Daytona has been repaved. The track was always faster than other tracks NASCAR raced at and several cars in a "train" were faster than single cars or smaller trains. This year is different, however, because two car groups are by far the faster than any other groupings. When a third car joins a tandem, all three slow down noticeably. Why is this? Why wouldn't it still be faster to have more cars pushing?</p>
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<p>So I have the following statement.</p>
<p>"A merry-go-round is spinning with a fixed angular speed. As a person is walking towards the edge, the force of static friction must increase in order for the person not to slide off."</p>
<p>I have no clue why is it true.
I tried to figure out the problem with the following equations:
rω = v
(mv^2)/r = Tension
but I can't seem to get past the tension. I guess what it causes the person to slide off would be the tangential acceleration (and therefore a friction force is needed), but I can't find a way of getting to this last variable with the previous mentioned equations...</p>
<p>I need help at figuring out specifically what (if I'm not right about the tangential acceleration) causes the person to slide off and how is it affected by an increase in the radius (as the person walk towards the edge.)</p>
| 8,095 |
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<p>What is the variation of <a href="http://en.wikipedia.org/wiki/Gauss%E2%80%93Bonnet_gravity" rel="nofollow">Gauss-Bonnet</a> term total derivative of?</p>
<p>i.e. Variation of Gauss-Bonnet combination $= \nabla_{\mu} C^{\mu}$. </p>
<p>What's $C^{\mu}$ in 4-dimensions?</p>
| 8,096 |
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] |
<p>I was recently introduced to <a href="http://en.wikipedia.org/wiki/Dimensional_analysis" rel="nofollow">dimensional analysis</a> and I wanted good references for learning the ideas behind it and representation of the natural world. I'm a grad student in biology. I don't have much of a physics background.</p>
| 8,097 |
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<p>Okay, so I know that the wavelength of an electron is <code>5e-7m</code> and I am asked to calculate its minimum velocity and hence minimum energy.</p>
<p>Calculating the minimum velocity via $\lambda = h/p$ gives <code>1454.67 m/s</code>. Next step is to calculate the associated energy.</p>
<p>The actual answer given to me is $E_{min} = 1/2 mv^2 = 9.71e-25J$, whereas I decided to calculate it using:</p>
<p>$E_{min} = hf = hv/\lambda_{max} = 1.93e-24J$. </p>
<p>Why does my answer differ?</p>
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<p>The headline question: Is it known how to construct an equivalent of the 3-D Ising Fixed point theory on an arbitrary 3-D manifold? Or any non-trivial d > 2 fixed point?</p>
<p>The answer is maybe as simple as a careful conformal mapping of the action . I don't even know whether this is a hard question not.</p>
<p>Let me try to explain what I mean by a fixed point theory on an arbitrary manifold and why I think such a thing should exist. By a fixed point theory I mean a QFT without a lattice scale, or put differently a theory with an actual continuum limit, with correlation function at all distances defined. If we have such a fixed point theory we can say it is the "Ising" if on small scales correlation functions look like the usual Ising correlation functions.</p>
<p>We know how to use RG to construct fixed point theories on the $d$ dimensional plane. But the usual technique relies on the Fourier transform, and is therefore specific to the plane (or maybe other symmetric spaces). </p>
<p>Does a fixed point theory exist for every manifold? It's true in $d = 2$, where I can use analytic maps to construct geometrically non-trivial theories. We can do it with topological QFTs. These are obviously special. It feels like scaling ideas should work - most crudely put a lattice on your manifold and then repeatedly refine the lattice while <em>locally</em> scaling the parameters, so that large scale correlation functions remain constant. But I don't exactly see how geometric information should be encoded in this process and one has a tremendous amount of freedom in local scaling. I haven't figured it out. Maybe the process forces you somehow to a trivial theory or a topological blow up.</p>
<p>Lastly, I haven't specified exactly what geometric information should go on this manifold. The naive assumption is that a conformal geometry is sufficient. But I honestly don't know.</p>
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<p><img src="http://i.stack.imgur.com/6EXpa.png" alt="enter image description here"></p>
<p>Why does the object not go inward, into the circle if the acceleration is inward? I think its because the velocity to outward? So they sort of cancel each other out? But if the speed is kept constant and the acceleration is inward, won't the object eventially move inward? </p>
<p>Also, what keeps the direction of acceleration always point inward? </p>
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<p>This is probably gonna be a dumb question but I don't know exactly where I am making the mistake. I have been taught in highschool that simple harmonic oscillator phase plot is the $sin(\omega t)$:
<img src="http://i.stack.imgur.com/v8EOE.png" alt="enter image description here"></p>
<p>However now that I am thinking about it, the phase is actually only the $\phi=2\pi f t$ part and therefore the real plot should look like this:
<img src="http://i.stack.imgur.com/syt8f.png" alt="enter image description here"></p>
<p>What is the problem with the second plot? Shouldn't the second one be the phase plot?</p>
<p>Note: Both of the plots are for the oscillator with $f=0.25$ and $t\in[0,10]$.</p>
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<p>Ok - fair warning - Non-physicist asking dumb-assed questions here again.
I've been reading a lot of Einstein, Feynmann, Ferris etc. I'm just loving this stuff.</p>
<p>But I suddenly found myself looking at the 4-D sphere concept of our expanding universe, as well as the distortion of spacetime caused by objects with mass and so on, and suddenly thought - this 4th spacial dimension (that we cannot comprehend) through which our universe is expanding - is this actually the dimension of time?</p>
<p>Is the time dilation experienced closer to bodies with mass not a literal curving of the skin of our 4-D sphere in toward its center? So at that point in spacetime the radius is increasing more slowly compared to the radius further from the mass?</p>
<p>I'm afraid my brain just keeps failing when I try to incorporate other concepts from GR into this view - such as time dilation with increasing speed. However the shortening of length of an object in the direction of travel kind of makes sense there as well but only if there was a similar unilateral distortion near bodies of mass.</p>
<p>And then I got to thinking about black holes and their extreme curvature - are they literally stopping universal expansion at their point of singularity such that deltaR=0 and therefore time stops? Or do they even stretch right back to R=0?</p>
<p>Ok, enough of my sillyness - can someone put this into perspective for me?</p>
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<p>The notes I'm using for QFT give an orthonormality condition for the following plane wave:
\begin{align*}
u_{r,k}(x) = \frac{1}{\sqrt{V}} \varepsilon_r(\mathbf{k})e^{ik \cdot x}
\end{align*}
(where $\varepsilon_r(\mathbf{k})$ is a linear polarisation vector satisfying $\varepsilon_r(\mathbf{k}) \cdot \varepsilon_s(\mathbf{k}) = \delta_{rs}$ and $\varepsilon_r(\mathbf{k}) \cdot \mathbf{k} = 0$.)</p>
<p>This condition is:
\begin{align*}
\int u_{r,k}(x) u_{s,k'}^*(x) d^3x = \delta_{k,k'}\delta_{r,s}
\end{align*}
So far so reasonable.</p>
<hr>
<p>Now, later on in the the notes I meet a similar plane wave:
\begin{align*}
u_k(x) = \frac{1}{\sqrt{V}} e^{i(k \cdot x - \omega_k t)} = \frac{1}{\sqrt{V}} e^{-ik_\mu x^\mu}
\end{align*}
I am trying to construct an othornomality condition, but this time with two positions, $x$ and $x'$:
\begin{align*}
\int u_k(x) u_{k'}(x')^* d^3x = \frac{1}{V}\int e^{i(k \cdot x - k' \cdot x')}e^{-i(\omega_k - \omega_{k'})t} d^3x
\end{align*}
Now I'm a little unsure about this. My gut says it becomes $\delta_{k,k'}\delta_{x,x'}$ but I am worried about the legitimacy of this, and overall I am confused about standard procedures for these kinds of integrals.</p>
| 8,103 |
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<p>In physics, usually <a href="http://en.wikipedia.org/wiki/Taylor_series" rel="nofollow">Taylor series</a> is used to express a quantity which keep changes with coordinate. For example the potential energy of a molecule changes with coordinate, so we express the potential energy as a Taylor expansion with respect to the coordinate about the equilibrium configuration. </p>
<p>I am confusing this with the <a href="http://mathworld.wolfram.com/MaclaurinSeries.html" rel="nofollow">Maclaurin series</a> in mathematics. If the equilibrium position is zero, it is Maclaurin series. We have the freedom to fix the equilibrium position at zero. In short, what is the physical significance of Maclaurin series? or else, what is the need of defining a Maclaurin series? Any physical interests on Maclaurin series?</p>
<p>One more question I would like to add. If a quantity is changing with respect to its coordinate and doesn't have an equilibrium at any of its configuration, then how we can express it? Taylor series can be applied only if the quantity is changing about an equilibrium configuration?</p>
| 8,104 |
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<p>My book uses the argument that the multiplicities of a few macrostates in a macroscopic object take up an extraordinarily large share of all possible microstates, such that even over the entire lifetime of the universe, if each microstate had an equal chance of being accessed, fluctuations away from equilibrium would never occur.</p>
<p>My question to this statistical proof is this: In the real world, is there really an infinitesimal but nonzero chance that macroscopic systems could access some of the more unlikely macrostates (e.g. heat transfer from a cold object to a hot object)?</p>
| 877 |
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<p>Temperature is the measure of movements of atoms. So if something is said to have high temperature it means that its atoms are moving fast or have high KE energy.</p>
<p>There are basically two ways heat can be travels from on object to another. 1) conduction. 2) Radiation. </p>
<p>My question is how does radiation causes increase in KE energy of atoms, when photons only interact with electrons? Or how does electrons going up in energy state translates to atom having more KE energy.</p>
| 8,105 |
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] |
<ol>
<li><p>Using high laser intensity to produce electron-positron pair, is it still required interaction with nucleus as is the case when gamma rays are used?</p></li>
<li><p>What causes the pair creation ? </p></li>
</ol>
| 8,106 |
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<p>I was recently wondering what would happen if the force sliding two surfaces against each other were somehow weaker than kinetic friction but stronger than static friction. Since the sliding force is greater than the maximum force of static friction, $F > f_s = \mu_s F_N$, it seems that the surfaces should slide. But on the other hand, if the force of kinetic friction is greater than the applied force, there'll be a net force $\mu_k F_N - F$ acting <em>against</em> the motion, suggesting that the surfaces should move opposite to the direction they're being pushed! That doesn't make sense.</p>
<p>The only logical resolution I can think of is that the coefficient of static friction can never be less than the coefficient of kinetic friction. Am I missing something?</p>
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<p>In principle, we could describe all physics without EM fields (or photons), as they are mainly a useful tool to describe "action at distance" (which does not mean instantaneous) between charged particle. In some sense, I could always integrate out photons, and describe only electrons and not have any changes in the observations (as detecting photons is done by observing how the motion of charged particles changes). With this picture, all photons are "virtual" (in the QFT sense), and we could expect them to always be off-shell.</p>
<p>My question is : why do we expect that photons that are "really emitted" (in a sense that should be made clearer, but that might mean "travelling very far before being absorbed") are always on-shell (i.e. having $E=p$)? Is it because the propagator of an off-shell photon decays very fast and therefore these photons can not interact with long distance charges ?</p>
<p>We could imagine that two very distant electrons (say, in two different galaxies) "scatter" each other (what we usually call "seeing a distant star") with off-shell photons. Why is it not so ? Is it just because the probability of this event is very small ?</p>
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<p>Consider the scattering of a quantum particle in one dimension, caused by a step in the potential (this appears in many undergrad level QM books):</p>
<p>$$ V(x) = \begin{cases} V_1 & x<0 \\ V_2 & x>0\end{cases}. $$</p>
<p>The particle is incident from the left, so it's wavefunction is:</p>
<p>$$ \psi(x) = \begin{cases} e^{i k_1 x} + r e^{-i k_1 x} & x<0 \\ t e^{i k_2 x} & x>0\end{cases}, $$</p>
<p>where $k_i =\sqrt{2m(E-V_i)}/\hbar$.<br>
Matching the wavefunction and its derivative at $x=0$ gives:</p>
<p>$$ r = \frac{k_1-k_2}{k_1+k_2} ~~~;~~~ t = \frac{2 k_1}{k_1+k_2}.$$</p>
<p>Now we put another step in the potential at some distance $L$, which makes it a box potential:</p>
<p>$$ V(x) = \begin{cases} V_1 & x<0 \\ V_2 & 0<x<L \\ V_1 & L<x\end{cases}. $$</p>
<p>We solve this in a similar manner as before, with the wavefunction:</p>
<p>$$ \psi(x) = \begin{cases} e^{i k_1 x} + r e^{-i k_1 x} & x<0 \\ a e^{i k_2 x} + b e^{-i k_2 x} & 0<x<L \\ t e^{i k_1 x} & L<x \end{cases}. $$</p>
<p>Matching the wavefunction and its derivative at $x=0,L$ gives:</p>
<p>$$ r = \frac{k_1^2-k_2^2}{k_1^2+k_2^2+2 i k_1 k_2 \cot{(k_2 L)} } ~~~;~~~ t = \text{(something)}.$$</p>
<p><strong>How come the second scattering problem doesn't reproduce the first scattering problem in the limit $L \rightarrow \infty$?</strong><br>
I'm looking only at the value of $r$. I send a particle in, it scatters, and I get something back with an amplitude $r$. It seems unphysical that if the potential changed at $x=L$, it changes the scattering at $x=0$, no matter how far $L$ is.</p>
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<p>How is a photon emission energy balance maintained ?</p>
<p>If an electron is in motion across space at 260,000 kilometers per second, and it releases a photon in that same direction of travel, the photon will be in motion across space at the speed of 300,000 kilometers per second, thus there will be a difference of 40,000 kilometers per second.</p>
<p>If another electron that is traveling along side the first electron at 260,000 kilometers per second, and it releases a photon in the reverse direction, the photon will be in motion across space at the speed of 300,000 kilometers per second, thus there will be a difference of 560,000 kilometers per second.</p>
<p>In both cases each electron will lose energy when releasing a photon. Is the loss of energy equal in both of these cases ? ( assuming each photon is is identical in its properties of course )</p>
| 8,110 |
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<p>I would like to know how this object would rotate in free space out of curiosity.</p>
<p>Below is a diagram of the object. It is of uniform density and has a center of mass in the center of the object, denoted by the green cross.</p>
<p>On each long end of the bar, on opposing sides, are thrusters. They are placed at exactly the same distance from the ends and from the center of mass. They provide the same amount of force. They are labeled "A" and "B" in the diagram, and the red arrows show which direction the thruster fires.</p>
<p>Edit: Assume that the thrusters are ideal and do not lose fuel/mass over time.</p>
<p><img src="http://i.stack.imgur.com/yZBqF.png" alt="Bar with thrusters"></p>
<p>If both thrusters are fired simultaneously, I believe the bar will spin about its center of mass, and there will be no translation.</p>
<p>My questions are about what happens when you fire only one of the thrusters.</p>
<h1>Question 1</h1>
<p>What happens if you fire only thruster B?</p>
<p>I think the bar will rotate, but not about it's center of mass, and not it's endpoints.</p>
<p>I think it will also translate, but not sure how.</p>
<h1>Question 2</h1>
<p>What equation(s) govern the motion of the body in this scenario? </p>
<p>I am almost certain that center of mass is a variable in this, because if you put a thruster directly in line with the center of mass, you would have no rotation at all, only translation.</p>
<h1>Question 3</h1>
<p>If left to spin to high velocities, will the object be "locked in" on it's translation vector?</p>
<p>Because it is spinning so fast, the thrust wouldn't have a significant net force in any particular direction, so the object would forever travel at some particular velocity and heading.</p>
<h1>Question 4</h1>
<p>What happens if you fire thruster B at 100% force, and thruster B at 50% force? </p>
<p>This question may already be answered by one of the other questions above.</p>
<p>Edit 2:
I wanted to see what this motion looked like, so I <a href="https://www.youtube.com/watch?v=QITGU6dw9xc" rel="nofollow">coded something up</a> in an attempt to model it. I only implemented angular and translational acceleration/velocity from the equations Physics Llama provided.</p>
| 8,111 |
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<blockquote>
<p><strong>Possible Duplicate:</strong><br>
<a href="http://physics.stackexchange.com/questions/19825/torque-required-for-a-motor-to-move-an-object-on-wheels">Torque Required For a Motor to Move an Object on Wheels?</a> </p>
</blockquote>
<p>I want to build a 4 wheel robot. the maximum weight of the robot is approximately <strong>100 kg</strong>. and the radius of my wheel is <strong>150 mm (outer dia 300 mm and inner dia 280 mm)**</strong> and the width of the wheel is <strong>120mm</strong>. I want to use <strong>4 motors in 4 wheel</strong> in order to run my robot. this robot will run on a very rough surface or the sand like surface. <strong>Can anyone pls tell me what is the required torque of every single motor?</strong> If I want to use 2 motor to run the robot then what will be the case? If anyone can suggest a formula for calculating the torque it will be helpful.</p>
| 267 |
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<p>This is a question about the properties and strength of glass. I know that glass is an amorphous solid but I am not sure if that is relevant to my question. My question is that if there's no physical cracks or blemishes, is it possible for glass to be damaged? For example, metal can be damaged by a sharp hit or force to it, as it is evident in a dent. For glass however, if a sharp force hits it and there is no evident crack or scratch, is the glass still in the same condition as it was before it got hit. For instance if I had something in contact with the glass that repeatedly struck it, would it over time weaken (ignore scratching)? (I know that with metal, each successive blow will slightly deform it)</p>
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<p>I'm looking at a review question I was given and it quite frankly has me stumped. </p>
<p>"Using matrix representations find $L^{3}_{x},L^{3}_{y},L^{3}_{z}$ and from these show that $L_{x}, L_{y},L_{z}$ satisfy the same algebraic equations."</p>
<p>What has me stumped is the $L^{3}_{x},L^{3}_{y},L^{3}_{z}$, I'm not even sure how it makes sense to talk of the cube of L? Any pointers in the right direction here would be incredibly helpful!</p>
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<p>I'm trying to develop my understanding of spinors. In quantum field theory I've learned that a spinor is a 4 component complex vector field on Minkowski space which transforms under the chiral representation of the Lorentz group. </p>
<p>Now I've been reading that we can derive spinor representations by looking at the universal covering group of the proper orthochronous Lorentz group, which is $SL(2,\mathbb{C})$. Now $SL(2,\mathbb{C})$ acts on $\mathbb{C}^2$ by the fundamental representation. My book (Costa and Fogli) then calls elements of $\mathbb{C}^2$ spinors.</p>
<p>But the second type of spinors have a different number of components to the first! What is going on here? Could someone clearly explain the link between these two concepts in a mathematically rigorous way? I come from a maths background of group theory and topology, but don't know much representation theory at present.</p>
<p>Many thanks in advance!</p>
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-0.027389096096158028,
-0.02471928298473358,
0.02343192882835865,
0.0003587176906876266,
-0.007639536168426275
] |
<p>A circular/inclined to the equator/non-polar orbit is typical for natural satellites; apparently a consequence of the energy/velocity requirements. This is applicable to natural satellites around planets, planets around Sol ... and perhaps upwards on the scale. </p>
<ul>
<li>When can a natural satellite possibly be in polar orbit around it's primary? </li>
<li>Are there any such observations on record?</li>
</ul>
| 8,115 |
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<p>I'm trying to solve the driven Two-Level System (TLS or qubit) question using a Fourier transform of the Schrodinger equation (SHE), but I'm getting stuck on solving the equation. </p>
<p>Given Hamiltonian
$$H=\left(
\begin{array}{cc}
-\frac{\omega _0}{2} & \frac{1}{2} V e^{i t \omega _D} \\
\frac{1}{2} V e^{-i t \omega _D} & \frac{\omega _0}{2} \\
\end{array}
\right)$$</p>
<p>and plugging into SHE:</p>
<p>$$i\frac{ d }{\text{dt}}\left(
\begin{array}{c}
c_a(t) \\
c_b(t) \\
\end{array}
\right)=H \left(
\begin{array}{c}
c_a(t) \\
c_b(t) \\
\end{array}
\right)$$</p>
<p>I get a set of two coupled 1st Order differential equations which i then Fourier transform and use the derivative rule and the shift rule to get:</p>
<p>\left(
\begin{array}{c}
-\omega c_a(\omega )=\frac{1}{2} V c_b\left(\omega -\omega _D\right)-\frac{1}{2} \omega _0 c_a(\omega ) \\
-\omega c_b(\omega )=\frac{1}{2} V c_a\left(\omega +\omega _D\right)+\frac{1}{2} \omega _0 c_b(\omega ) \\
\end{array}
\right)</p>
<p>If I shift the $c_b$ equation:</p>
<p>$$-\omega c_b\left(\omega -\omega _D\right)=\frac{1}{2} V c_a\left(\omega _D-\omega _D+\omega \right)++\frac{1}{2} \omega _0 c_b\left(\omega -\omega _D\right)$$</p>
<p>and solve for term i can plug back into the first equation:
$$c_b\left(\omega -\omega _D\right)=\frac{V c_a(\omega )}{2 \omega +\omega _0}$$</p>
<p>and solve for $c_a$:</p>
<p>$$c_a(\omega )=\frac{V^2 c_a(\omega )}{(2 \omega )^2+\omega _0^2} $$</p>
<p>So it looks like a lorentzian function with a width of the resonance frequency and centered at 0 multiplies $c_a$, but $c_a$ cancels unless it is 0? or some delta function?</p>
<p>This is where I do not understand how to proceed to solve the problem further?
Is part of the challenge that I am solving the problem without boundary conditions, just trying to find the steady state?</p>
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<p>Consider two orbits $x(t),\space y(t)$ representing the origin and destination for some spaceflight of interest. These could be, for example, cycloids describing LEO and another orbit circling, say, the moon or Mars.</p>
<p>Suppose I want to leave at $x(t_i)$ and arrive at $y(t_f)$, departing and joining those orbits 'smoothly', so their derivatives $\dot{x}(t_t)$ and $\dot{y}(t_i)$ are also imposing boundary conditions on my trajectory.</p>
<p>To accomplish this, I allow for rocket engines to be fired, most notably while leaving one orbit and decelerating into the other, but also at any time mid-flight. On the other hand, I restrict the total spent fuel along the trip, or the total "delta-v" changes, or some equivalent. </p>
<p>What would be the best way to solve for this trajectory, including the firing pattern of the engine?</p>
<p>What I thought about was devising some adjusted Lagrangian that includes the impulse caused by the engines and adding to it a Lagrange multiplier imposing fuel consumption. This is then used to derive new equations of motion. For example, I would consider an action such as:
$$S=\int_{t_i}^{t_f}L[q,\dot{q},F_e,t]dt+\lambda\left(J-\int_{t_i}^{t_f}|F_e(t)|dt \right)$$
Where $q(t)$ is the desired orbit, $F_e(t)$ is the force exerted by the engines and unknown <em>a priori</em>, and $J$ the total impulse supplied by them along the journey$^\dagger$. Now derive EOMs and solve for $q,F_e$.</p>
<p>Eventually, I'd like to explore if this 'formalism' produces solutions (trajectories) that can be inverted, solving either for $J$ (allowing minimization of fuel) or $t_f,t_i$ (minimizing flight time, picking launch window).</p>
<p>Where I am struggling is how to modify said Lagrangian. Should $F_e$ be considered as another generalized coordinate? If so, then a kinetic term for it is needed. Should it be treated as another term in the potential energy? Maybe it should be restated as a force originating from enforcing some constrained orbit? Maybe it's actually another 'Lagrangianesque' function of $q, \dot{q}$ and perhaps $\ddot{q}$ as well?</p>
<p>$^\dagger$Admittedly, the instantaneous thrust is not constrained, but it's a start.</p>
<p><strong>EDIT</strong></p>
<p>Following the reference supplied in the answer below, I managed to make some progress with the problem. This was done by "inverting roles", so now the "Lagrangian" is the engine force and a Lagrange multiplier enforces the Newtonian equation of motion:</p>
<p>$$S=\int_{t_0}^{t_f}dt\left( F^2 + 2\lambda(t)[m\ddot{x}+m \nabla \phi-F] \right)$$</p>
<p>Now, taking the differential of $S$ with regards to $x, F$ and $\lambda$, and doing the routine integration by parts on the variation $\delta\ddot{x}$, we obtain the following equations of motion:</p>
<p>$$(1) \space F=\lambda$$</p>
<p>$$(2) \space m\ddot{x} = -m\nabla\phi +F$$</p>
<p>$$(3) \space \ddot{\lambda} = - H_{\phi}*\lambda$$</p>
<p>Where $H_{\phi}$ denotes the Hessian matrix of $\phi$, i.e. $(H_{\phi})_{ij}=\partial_i\partial_j\phi$.</p>
<p>These equations can know be solved with given boundary conditions $x(t_0),x(t_f),\dot{x}(t_0),\dot{x}(t_f)$, which proved to be pretty straight-forward to do in MATLAB.</p>
<p>The next thing on the agenda is making $t_0, t_f$ free variable subject to optimization as well, so we can find the best launch window. This is proving to be a bit more difficult. Introducing these extra degrees of freedom requires more Lagrange multipliers and produces algebraic equations for each of them and the multipliers. The major difficulty is that the equations of motion above can't be easily integrated and then solved for the time $t$, so they can be evaluated at the launch and landing times. Any suggestions?</p>
| 8,117 |
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<p>If one measures the projection of spin of a spin half particle along the $x$ axis one will always get $\pm\tfrac12\hbar$.</p>
<p>Measuring it along the $y$ axis one will always get $\pm\tfrac12\hbar$.</p>
<p>Measuring it along the $z$ axis one will always get $\pm\tfrac12\hbar$.</p>
<p>Measuring it along "any" axis one will always get $\pm\tfrac12\hbar$.</p>
<p>Why are the possible outcomes the same in all directions?</p>
| 8,118 |
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<p>I read from some books of number theory that $$\sum_{n=1}^{\infty}\frac{1}{n^s} = -\frac{1}{12}\text{,when } s=-1.$$</p>
<p>Now is there such a result $$\sum_{n=1}^{\infty}\frac{1}{n^s} = \pi \text{,when } s=1,$$or $$\sum_{n=1}^{\infty}\frac{1}{n^s} = c \pi \text{,when } s=1 \text{,where } c \text{ is a rational number ?}$$ </p>
<p>I get a similar result in mathematics by analogue, I suspect the result may have some interpretation in physics.</p>
| 8,119 |
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<p>I've been wondering, what causes the normal force to exist?</p>
<p>In class the teacher never actually explains it, he just says "It has to be there because something has to counter gravity." While I understand this is true, it never explains <i>why</i>.
Whenever I ask anyone else they always respond in a similar way, saying "It has to be there, because the object is not accelerating", and this has become very frustrating.</p>
<p>So what is the cause of the normal force? From my reasoning, it has to be one of the four fundamental forces. (Gravity, electromagnetism, the weak force, or the strong force). It would seam to me that electromagnetism would make to most sense (electrons in the outer shells of atoms repelling each other), </p>
<p>However, just as I thought this had to be right, I read a thing online about "certain fundamental particles repelling each other when their wave functions overlap". I haven't studied quantum mechanics yet so I'm not really sure what to make of that.</p>
<p>If anyone could shed some light on this for me it would be much appreciated.</p>
| 268 |
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<p>I would like to know how much of the Big Bang theory is dependent on the constancy of the speed of light. </p>
<p><strong>P.S.</strong>: It might be guessed that I am asking this because of the recent CERN news.
Yes, of course, I am. I was having some chat over this topic and I hit a wall before I could come to an answer for this because of my not-more-than-school-level-Physics knowledge.</p>
<p>Thank you.</p>
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<p>The <a href="http://en.wikipedia.org/wiki/Crystallography_restriction" rel="nofollow">crystallography restriction</a> states that any 2-dimensional lattice can have rotational symmetry of degree 1, 2, 3, 4 or 6 - and that's it. A simple proof of I've heard of this is: the magnitude of $rot_\theta(x)-x$ is less than the magnitude of $x$ if $\theta$ is less than $\pi / 3$. (So if there's a rotation by less than $\pi/3$, it will fail to be a lattice, since we can always get smaller and smaller elements.)</p>
<p>That sounds fine to me, and I get why real crystals would have the requirement that all atoms be some epsilon away from each other. I don't understand why nature requires the lattice to be closed though - obviously real crystals aren't completely closed under addition, or else they would be infinitely large.</p>
<p>So why does the crystallography restriction limit the symmetries of real crystals? (Or does it not, and this application of the theorem is made up by math teachers to convince their students that math is useful?) </p>
| 8,121 |
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<p>The weight function comes from Dirac's book, PRINCIPLES OF QUANTUM MECHANICS. On page 66,he says that sometimes it is more convenient not to normalise the eigenvectors, i.e. $$\langle\xi_1'...\xi_u'|\xi_1''...\xi_u''\rangle=\rho'^{-1}\delta(\xi_1'-\xi_1'')...\delta(\xi_u'-\xi_u'')$$</p>
<p>Thus we have $$\iiint|\xi_1'...\xi_u'\rangle\langle\xi_1'...\xi_u'|\rho'd\xi_1'...d\xi_u'=1$$</p>
<p>Here Dirac says that $\rho'd\xi_1'...d\xi_u'$ is the weight attached to the volume element of the space of the variables $\xi_1',...,\xi_u'$.</p>
<p>If we consider the space of the variables $\xi_1',...,\xi_u'$ as a Riemann manifold $M$ with metric $g$, then the volume element of $(M,g)$ is $\sqrt{\det(g)}d\xi_1'...d\xi_u'$. Does the two volume element equal? i.e. do we have that $\rho'=\sqrt{\det(g)}$?</p>
<p>In his book,he then gives an example where the $\xi$'s are the polar and azimuthal angles $\theta,\phi$ giving a direction in three-dimensional space. In this case we have $\rho'=\sin{\theta'}$. And if we calculate the $\sqrt{\det(g)}$ of the unit sphere(in coordinates $\theta,\phi$),it is $\sin{\theta'}$,too!</p>
<p>Is it just coincidence? Or is there an explanation?</p>
<p>Would someone be kind enough to give me some hints on this? Thank you very much!</p>
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0.022795328870415688
] |
<p>I just read <a href="http://idealab.talkingpointsmemo.com/2011/09/mit-setting-up-industrial-scale-graphene-printing-press.php" rel="nofollow">this story</a> on MIT working on industrial scale, km^2 sheet production of graphene.</p>
<p>A quick check of <a href="http://en.wikipedia.org/wiki/Graphene#Mechanical_properties" rel="nofollow">Wikipedia on graphene</a> and <a href="http://en.wikipedia.org/wiki/Space_elevator#21st_century" rel="nofollow">Wikipedia on space elevator</a> tells me</p>
<blockquote>
<p>Measurements have shown that graphene has a breaking strength 200 times greater than steel, with a tensile strength of <strong>130 GPa</strong> (19,000,000 psi)</p>
</blockquote>
<p>and</p>
<blockquote>
<p>The largest holdup to Edwards' proposed design is the technological limit of the tether material. His calculations call for a fiber composed of epoxy-bonded carbon nanotubes with a minimal tensile strength of <strong>130 GPa</strong> (19 million psi) (including a safety factor of 2)</p>
</blockquote>
<p>Does this mean we may soon actually have the material for a space elevator?</p>
| 8,123 |
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<p>Googling yields contradictory results, so here my question:</p>
<p>When I sing, my vocal chords vibrate, but my whole body is the resonance body, right? So I would say that when I think about standing waves etc. I must think of a resonantor that's closed (i.e. hard boundary conditions) at both ends, because my bottom end is certainly closed, and the top end has the vocal chords which are vibrating similar to woodwind instruments, so it's closed on that end as well.</p>
<p>If I am mistaken, could someone point out the physics behind this?</p>
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<p>How much energy would it take to add to the earth's orbital velocity to push the earth into mars' orbit?</p>
<p>My thoughts:
So, in 1-2 billion years, the sun will gradually get brighter.</p>
<p>Suppose a future civilization wanted to gradually move the earth farther from the sun. And for, now, please ignore that Mars is already in Mars' orbit. let's pretend there is room there.</p>
<p>I'm wondering what the answer looks like, in power terms, over the course of a billion years? are we talking more power than modern civilization, or if we have a billion years can we do it with just a small generator's worth of power output?</p>
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<p>I have been doing some readings on superconductivity and have come across the <a href="http://en.wikipedia.org/wiki/London_penetration_depth" rel="nofollow">London penetration depth</a>. I somewhat understand how it comes into play with the London equation and Ampère's law.</p>
<p>Right now, I am having a hard time finding out how one would, if one is given a material that is said to be a superconductor and knowing only the critical temperature, be able to determine the $\lambda_L$ of this material.</p>
<p>I know that it is possible to use <a href="http://en.wikipedia.org/wiki/Muon_spin_spectroscopy" rel="nofollow">μSR</a>, but how exactly would one go about doing that? Also, is that the only experimental method if one does not know anything else about the material?</p>
| 8,126 |
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<p>I understand that assuming light is quantized implies that if a lower energy interacts with metal, it is possible that that quanta will not have enough energy to eject the electron. What prevents two quanta from interacting with the electron at the same time, though, and summing to have enough energy to release the particle. Indeed, what if two separate light rays of the same frequency impact the electron, why can't that free it? </p>
| 388 |
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] |
<p>I'm trying to track down the origin of the False Vacuum paradigm. Is it Coleman and de Luccia's "Gravitational Effects on and of Vacuum Decay"
<a href="http://www.slac.stanford.edu/pubs/slacpubs/2000/slac-pub-2463.html" rel="nofollow">http://www.slac.stanford.edu/pubs/slacpubs/2000/slac-pub-2463.html</a></p>
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<p>I heard that Schrodinger pointed out that (classical/statistical) thermodynamics is impaired by logical inconsistencies and conceptual ambiguities.</p>
<p>I am not sure why he said this and what he is talking about. Can anyone point some direction to study what he said?</p>
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<p>Higgs Boson detected at LHC is massive. It has high relativistic mass means it has non-zero rest mass.</p>
<p>Higgs Boson gives other things rest mass. But, how does it get rest mass by itself?</p>
| 269 |
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<p>I'm trying to figure out a nice way to describe to a kid the physics behind these experiments:</p>
<p>Assuming ideal conditions, we have a small boat with a sale, close to a lake's shore and a fan fixed on the shore, oriented towards the boat's sale. Once the fan is turned on, the boat will start to move slowly away from the lake's shore.</p>
<p>Now, let's replace the fan with a small kid having a long wooden rod in his hand and let's set the boat back in the initial position. Once the kid touches the boat's sail with the rod, the boat will start moving in the opposite direction, slowly...</p>
<p>In the next experiment, the kid notices that his rod is not long enough to propel the boat further away, so he decides to climb in the boat and to try the same thing he did while he was on the shore: push the sail gently with the wooden rod, but, surprise: the boat did not budge an inch.</p>
<p>The kid thought that maybe the rigid nature of the wooden rod is to blame for the failure of the experiment, so he decided to try again, this time by placing the fan on the boat, oriented towards the sail. Did the boat start to move? Why?</p>
<p>Now, I could try to apply <a href="http://en.wikipedia.org/wiki/Momentum" rel="nofollow">linear momentum</a> theory and consider the air as being formed by small beads, each having an impulse $ p_i = m_i v_i $, where the masses and the speeds of the particles are roughly equal, and each $ p_i $ gets transferred to the sail, but I'm not sure if this gives any reasonable intuition to an elementary school kid.</p>
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<ol>
<li><p>Is it <strong>always</strong> allowed to represent the <a href="http://en.wikipedia.org/wiki/Metric_tensor" rel="nofollow">metric tensor</a> $g_{\mu \nu}$ in <a href="http://en.wikipedia.org/wiki/General_Relativity" rel="nofollow">General Relativity</a> as a $4\times 4$ matrix?</p></li>
<li><p>If the last one is represented for example with a $4\times 4$ matrix $\mathbf{A}\in{\mathbb{R}^{4\times 4}}$ is it always true that its metric inverse $g^{\mu \nu}$ is the <strong>ordinary</strong> matrix inverse $\mathbf{A}^{-1}$ which one can calculate with ordinary linear algebra methods?</p></li>
</ol>
<p>Maybe this question may sound silly, but I've understood that in differential geometry, also simple things become complicated...</p>
| 8,130 |
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<p>The <a href="http://en.wikipedia.org/wiki/Shell_theorem" rel="nofollow">shell theorem</a> states that:</p>
<blockquote>
<p><em>A uniform spherical shell of matter attracts a particle that is outside the shell as if all the shells mass were concentrated at its center.</em></p>
</blockquote>
<p>Now, I have to solve a problem related to this theorem. The statement of the problem is:</p>
<blockquote>
<p><em>A particle is to be placed, in turn, outside four objects, each of mass $m$:</em></p>
<p><em>(1) a large uniform solid sphere,</em> </p>
<p><em>(2) a large uniform spherical shell,</em> </p>
<p><em>(3) a small uniform solid sphere, and</em></p>
<p><em>(4) a small uniform shell.</em> </p>
<p><em>In each situation, the distance between the particle and the center of the object is $d$. Rank the objects according to the magnitude of the gravitational force they exert on the particle, greatest first.</em></p>
</blockquote>
<p>As far as I understand this problem (or maybe I've not understood it yet) I think all of the objects should be ranked first. This is because, for the large uniform solid sphere if we imagine that its entire mass $m$ is concentrated at its center which is at a distance $d$ from the particle, then it will attract the particle with the magnitude $\frac{Gmm_0}{d^2}$ where $m_0$ is the mass of the particle. And, each of the other three objects' centers distance is $d$ from the particle, and each of their masses is $m$, so it seems to be the case that all of them gravitate the particle with the same magnitude. </p>
<p>Do the objects gravitate the particle with different magnitudes, and hence should be ranked differently? What is the reason behind it if so is the case?</p>
| 8,131 |
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<p>Black holes have no-hair so there are uniquely specified by a mass, charge and angular momentum. Imagine a cloud of charged rotating dust. There will be a magnetic field associated with the current of charge that is rotating. As this cloud collapses to form a black hole, how is the magnetic field excluded from the region of the black hole?</p>
<hr>
<p>These three questions are similar but I think the answers will be different for each one:</p>
<blockquote>
<p><a href="http://physics.stackexchange.com/questions/39161/what-happens-to-an-embedded-magnetic-field-when-a-black-hole-is-formed-from-rota">What happens to an embedded magnetic field when a black hole is formed from rotating charged dust?</a>
It seems to me a rotating charged black hole must have a dipole
magnetic field. But the strength of the dipole field seems like an
extra parameter that black holes are forbidden by the no-hair theorem.</p>
<p><a href="http://physics.stackexchange.com/questions/39174/if-a-magnetic-monopole-falls-into-a-schwarzchild-black-hole-what-happens-to-the">If a magnetic monopole falls into a schwarzchild black hole, what happens to the magnetic field?</a>
Here there would be only radial magnetic field lines leaving from the
event horizon to infinity. So if magnetic charge is counted as charge
this should be no problem. But if the black hole were rotating
wouldn't that produce an electric dipole field?</p>
<p><a href="http://physics.stackexchange.com/questions/39173/when-a-neutral-star-with-a-magnetic-field-collapses-to-form-a-black-hole-what-h">When a neutral star with a magnetic field collapses to form a black hole, what happens to the magnetic field?</a>
Here there is no charge so how can there be a magnetic field
associated with a black hole? That would definitely violate the
no-hair theorem.</p>
</blockquote>
| 403 |
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<p>I am a little stuck on the concept involved in part b) of the following problem:</p>
<p>A $5 m$ long road gate has a mass of $m = 15 kg$. The mass is evenly distributed in the gate. The gate is supported at point A by a rod attached to the ground below the gate. Point A is located $20 cm$ from the left end. The gate rotates around a point B located $80 cm$ from the right end, also supported by a rod attached to the ground. At the right end of the gate, a block of mass $25 kg$ is placed. The block's mass center is $15 cm$ from the right end.</p>
<p>a) Find the forces at A and B</p>
<p>b) How much force must be applied downward at the right end of the gate in order to swing the gate open?</p>
<p>OK, so for part a) I simply used the fact that since we are in equilibrium we must have zero net forces and zero net torque. This gave me $F_A = 23 N$ and $F_B = 369 N$, which is correct. In part b), I know the answer is supposed to be $113 N$, and I have been told that in order to find this I have to find out what force I have to apply to the gate in order to get $F_A = 0$. But this is a bit conceptually confusing to me. If the system is in equilibrium initially, wouldn't then just the slightest bit of force added cause the system to lose its equilibrium and start moving? </p>
<p>If anyone can explain this to me, I would be very grateful.</p>
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<p>I don't really understand horizontal projectile motion.
Suppose I throw a ball horizontally.
Given $\theta$ = 0 degrees, time = 0.5 seconds, horizontal range = 15.5 metres, </p>
<p>the "initial horizontal velocity of the ball" would just be the average velocity which is displacement divided by time? Or is it $V\sin\theta$? </p>
<p>Also, what would the "final velocity" be? It would be $\sqrt{V_x^2 + V_y^2}$ using the final values since $\theta$ changes?</p>
<p>And would I say that the maximum height reached by the ball is 0 metres or -1.2 metres? </p>
<p>Finally, in order for me to calculate Kinetic Energy, Work, Force, would I need to measure the mass of the ball or can I get the mass using the information I already know such as velocity and displacement and time? </p>
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<p>Suppose there is a perfect ideal conducting solid sphere. Suppose somehow a charge of $+Q$ is kept exactly at the center of the sphere and its surface is also given a $+Q$ charge uniformly distributed at the same time.</p>
<p>We know electric field inside a charged conductor is zero. But in this situation, a $+Q$ charge has appeared exactly at the center of the charged conductor. Will the electric field still be zero? If there is electric field inside,since net force on the center charge is zero and since the surface charges cannot go outside the sphere, the situation is like holding the surface charges with rigid sticks from the center, can a situation like this be created or will the center charge also go to the surface??</p>
<p>Lets take an electron and put it at the center and many electrons uniformly distributed on the surface of the sphere at the same time. Now net force on the electron is zero. Will it stay there and if we move it slightly it will come to the surface or will it come to surface anyway?</p>
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] |
<p>A week ago I <a href="http://physics.stackexchange.com/q/38963/2451">asked</a> people on this forum what mathematical background was needed for understanding Quantum Physics, and most of you mentioned Linear Algebra, so I decided to conduct a self-study of Linear Algebra. Of course, I'm just 1 week in, but I have some questions. </p>
<ol>
<li><p>How is this going to be applicable to quantum physics? I have learned about matrices (addition, subtraction, multiplication and inversion) and about how to solve multiple equations with 3 unknowns using matrices, and now I am starting to learn about vectors. I am just 1 week in, so this is probably not even the tip of the iceberg, but I want to know how this is going to help me. </p></li>
<li><p>Also, say I master Linear Algebra in general in half a year (I'm in high school but I'm extremely fast with maths), what other 'types' of math would I need to self-study before being able to understand rudimentary quantum physics mathematically? </p></li>
</ol>
| 53 |
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] |
<p>Where does a sparkly (CO$_2$) water bottle have more pressure? At a higher altitude (7000 feet) or under sea level (-8 feet)?</p>
| 8,135 |
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<p>I watched this Google Tech Talk recently - <a href="http://www.youtube.com/watch?v=dEaecUuEqfc" rel="nofollow">Quantum Conspiracy</a></p>
<p>Now I have done some investigating concerning these ideas and their implications, and also have dug up two relevant science papers written on this subject. You'll have to look them up on Google as I cannot yet post multiple links: </p>
<p>The Ithaca Interpretation of Quantum Mechanics </p>
<p>& </p>
<p>Quantum Information Theory of Entanglement and Measurement </p>
<p>What I would really like to know is if this theory is at all plausible, and if Ron in the presentation above is correct would this in any way imply that time is static in the Multiverse in that it is timeless? </p>
<p>I guess this has more to do with relativity than with quantum mechanics, but this wave function I take it according to the video and papers is a four-dimensional thing. One of those dimensions emerges as “time” in our classical universe(s), but which dimension that is depends on your frame of reference. A classical universe is a little slice of the wave function that contains macroscopic systems in classical correlation with each other across all four dimensions (as described by QIT). It is these correlations that give the illusion that time is passing. </p>
<p>This is covered briefly(in the talk above) at the end of the talk where he points out that the physical mechanism that produces Einstein’s “spooky action at a distance” also produces the “spooky action across time” that causes the past to appear as a coherent consistent whole, but to recap: If you and I look at something and agree that it is green, it is tempting to think that the reason we agree on this is that the thing we are looking at does in fact exist and is in fact green. And for macroscopic things that’s a pretty good approximation. But for particles it’s not. The reason you and I can look at a photon and agree that it’s vertically polarized is NOT that the photon in fact exists and is in fact vertically polarized, it’s because you and I comprise a macroscopic system of mutually entangled particles (by virtue of having interacted with the same photon) and so, by QIT, when we trace over (i.e. ignore) the photon we get a mathematical description of us as two classically correlated systems.</p>
<p>The same thing happens to “you” at different instances in “time". You are not in fact a physical being, you are a macroscopic system of mutually entangled particles (or, to be strictly correct, mutually entangled components of the wave function) which, as described by QIT, behave as if they were a set of classically correlated systems. The reason you at any instant in time feel like a material being that has existed in the past is because you have memories, which are part of your (classical) state, and those memories correlate with other things in the universe, like the pictures in your photo album. These correlations that persist across time are the reason that you believe the past to be real (as opposed to, for example, your dreams which don’t persist across time in the same way). Again, it is tempting to believe that the past is coherent because the things that you remember existing in the past did in fact exist in the past. But that is not the case. The reason the past is coherent is, again, because “you” at any instant in “time" are in classical correlation with parts of the wave function that “you” call “the past”. That is the <em>definition</em> of the past. That is the <em>reason</em> that time appears to flow in one direction despite the fact that all the laws of physics are time-symmetric. The “past” is simply that part of the wave function that “you” are correlated with at any particular instant (or, to phrase it more accurately, that any particular you is correlated with).</p>
<p>The simulation(Multiverse) cannot “go on forever” because it does not “go on” at all. The wave function is static. Time is an illusion that emerges when you slice up the wave function in a particular way that yields subsets of itself that behave like classically correlated systems across four dimensions. But these systems do not have the same metaphysical status as the wave function itself, so it is a sort of “category error” to try to apply words from one (like “beginning” and “end”) to the other.</p>
<p>My problem with this is that I always interpreted the Multiverse(if one even exists) as predicted by string theory to be eternal in that it doesn't require a cause and that time has no beginning and no end. However time cannot be static in this Multiverse because nothing would ever happen or everything would happen at once. In that sense inflation and time within each Universe would not make sense within the Multiverse and would be non-observable within the Multiverse. This doesn't make any sense to me at all. So time cannot be static in the Multiverse or linear like an arrow. So something doesn't add up with QIT.</p>
<p>Any support or clarification here from a real physicist or someone with more physics knowledge than myself would be greatly appreciated.</p>
<p>Thanks in advance</p>
| 8,136 |
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<p>I am given the fact that at a certain pressure a liquid boils at a corresponding temperature, at a different pressure it boils at a different temperature, and then I am asked to find the latent heat from that information, how do I do that?</p>
| 8,137 |
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<p>When you tune your radio (digital or analog) to receive say 100 MHz frequency and while in the environment there are hundreds of channels everywhere around the radio. How does it chooses to receive the frequency only at 100 MHz? How does the radio receiver works? Is it possible to explain this in simple concept?</p>
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<p>I was trying to understand the derivation of the boundary layer equations at p.145 of <a href="http://www.unimasr.net/ums/upload/files/2012/Sep/UniMasr.com_919e27ecea47b46d74dd7e268097b653.pdf" rel="nofollow">http://www.unimasr.net/ums/upload/files/2012/Sep/UniMasr.com_919e27ecea47b46d74dd7e268097b653.pdf</a>. : the derivation is completely given at that page. </p>
<p>I almost figured it all out, but I don't understand (12-8). With the approximations they made, I should get (12-8), but not the last term $\mu\frac{\partial^2v_y}{\partial x^2}$. I don't see how they get this term, when approximating $\tau_{xy} = \mu \frac{\partial^2 v_y}{\partial x^2}$ and $\sigma_{yy} = -P$ does not give the result shown. I wonder if I'm wrong, or the book is just mistaken? I hope someone can help me out here.</p>
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<p>In simple terms, why is beta negative decay more common than beta positive? </p>
<p>I know it's something to do with occuring inside/outside the nucleus - but I can't find a simple, easy to understand explanation!</p>
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<p>Back in college I remember coming across a few books in the physics library by <a href="http://en.wikipedia.org/wiki/Mendel_Sachs" rel="nofollow">Mendel Sachs</a>. Examples are:</p>
<p><em>General Relativity and Matter</em></p>
<p><em>Quantum Mechanics and Gravity</em></p>
<p><em>Quantum Mechanics from General Relativity</em></p>
<p><a href="http://arxiv.org/abs/1010.3630" rel="nofollow">Here</a> is something on the arXiv involving some of his work.</p>
<p>In these books (which I note are also strangely available in most physics department libraries) he describes a program involving re-casting GR using quaternions. He does things that seem remarkable like deriving QM as a low-energy limit of GR. I don't have the GR background to unequivocally verify or reject his work, but this guy has been around for decades, and I have never found any paper or article that seriously "debunks" any of his work. It just seems like he is ignored. Are there glaring holes in his work? Is he just a complete <a href="http://en.wikipedia.org/wiki/Crackpot_theory" rel="nofollow">crackpot</a>? What is the deal?</p>
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<p>This is a relativity-paradox which I can't resolve:</p>
<p>The distance between interference lines in the double-slit experiment is:</p>
<p>$$\ w = z \lambda/d$$</p>
<p>Where:</p>
<p>w: Distance between fringes</p>
<p>z: Distance from slits to screen.</p>
<p>$\lambda$: Wave length of light</p>
<p>$d$: Distance between slits.</p>
<p>Now, looking at the setup from a reference frame traveling in the direction of the distance $z$ between slit and screen and moving away from the slit in direction of the screen, the light gets redshifted and the distance $z$ gets Lorentz contracted:</p>
<p>$$\ w' = \frac{z}{\gamma} \gamma(1+\frac{v}{c}) \lambda/d =z(1+\frac{v}{c}) \lambda/d$$</p>
<p>This only holds for the first fringe or so, that aren't too far from the center, because the red-shift formula changes farther out. However, it's obvious that the distance between fringes has increased!</p>
<p>But vertical distances shouldn't change when boosting: I could place a photodetector at the first black fringe which beeps if it gets hit by a photon. In the reference frame at rest, it doesn't beep. But when looked at from the boosted reference frame, the fringes move and light hits the photodetector. It beeps. (The photodetector will not change its vertical position in a boosted frame - Lorentz contraction only happens along the axis of movement)</p>
<p>Where is my mistake?</p>
| 8,142 |
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<p>I have to prove the (non abelian) gauge invariance of the following lagrangian (for a certain value of $\lambda$):
$$\mathcal L= -\frac14 F^{\mu\nu}_aF_{\mu\nu}^a + \frac{k}{4\pi}\epsilon^{\mu\nu\rho}\operatorname{tr}[A_{\rho}\partial_{\mu}A_{\nu} + \lambda A_{\mu}A_{\nu}A_{\rho}]$$
Is there an easier way than expliciting every component of A, going to first order gauge transformation, ... ? Cause it seems really ugly to me.</p>
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<p>As photons bounce around and finally find their way into our eyes, are they continually relinquishing the information of the previous thing they "bounced off of"? Is this the reason why we receive a coherent, continuous image and not a chronologically chaotic image of the past meshed with the present? </p>
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<p>A ball is launched as a projectile with initial speed $v$ at an angle $\theta$ above the horizontal. Using conservation of energy, find the maximum height $h_\text{max}$ of the ball's flight.
Express your answer in terms of $v$, $g$, and $\theta$.</p>
<p>I am doing:</p>
<p>$$\begin{align*}
\frac{1}{2}mv^2 &= mgh_\text{max} + \frac{1}{2}m(v\cos\theta)^2\\
v^2 &= 2gh_\text{max} + (v\cos\theta)^2 \\
h_\text{max} &= \bigl(v^2 - (v\cos\theta)^2\bigr)/2g \\
h_\text{max} &= v^2\bigl(1 - (\cos\theta)^2\bigr)/2g \\
h_\text{max} &= \frac{v^2\sin^2\theta}{2g}
\end{align*}$$</p>
<p>Is that correct, or is there a much easier way...?</p>
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<p>This might be a question purely of words and the meaning of them but isn't Newtonian physics more "relative" than Einstein's Special relativity? Newtonian physics predicts that laws of momentum & inertia behave the same at whatever speed you are traveling. Einstein said things change, and do more and more the closer you get to the speed of $c$. You become more and more massive as you approach $c$, therefore requiring more and more energy until you would require infinite energy to break the speed of light.</p>
<p>Why is Einstein's theory called the theory of relativity when it described a world where (for the first time) physics occurring in one spacecraft traveling fast is not analogous to the physics in another spacecraft standing still?</p>
<p>Apologies if this is a stupid question.</p>
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<p>I have exactly 0 clue on how to start this problem, but I would be forever grateful for a hint in the right direction.</p>
<p>Given the operators $\hat x=x$ and $\hat p=-i\hbar \frac{d}{dx}$, prove the following relation:</p>
<p>$$
[\hat x, g(\hat p)]=i\hbar\frac{dg}{d\hat p}.
$$</p>
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<p>I'm working on a project and I need help generating a strong magnetic field over an air gap with the lowest material weight that could be achieved. I have tried to use permanent magnets but its not effective so I need to know the best way to generate a strong fixed electromagnetic field with the lightest material.
The attached image is a simple drawing shows the direction I need to generate the field in the boxes are the field source.</p>
<p><img src="http://i.stack.imgur.com/pPDLy.jpg" alt="enter image description here"></p>
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<p>Why in the general case of classical field theory canonical stress-energy tensor doesn't have symmetry of the permutation of the indices? </p>
<p>For explanation, let's have a "derivation" of an expression for tensor by using symmetry of translations.</p>
<p>For translations we have
$$
x'^{\nu} = x^{\nu} + \omega^{\nu } \Rightarrow X^{\nu}_{\alpha } = \delta^{\nu}_{\alpha}, \quad Y_{k, \alpha} = 0,
$$
so Noether current has following expression,
$$
J^{\mu}_{\alpha} = -\left(\frac{\partial L}{\partial (\partial_{\mu}\Psi_{k})}\partial_{\alpha} \Psi_{k} - \delta^{\mu}_{\alpha}L\right),
$$
and it isn't symmetrized.</p>
<p>Does it have some fundamental meaning?</p>
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