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<p>Schrödinger’s cat, the thought experiment, makes it seem like as if measurement can cause a system to stop being in a superposition of states and become either one of the states (collapsed).</p>
<p>So does a system always exist in a superposition? In this sense, do things in the macroscopic world have a wavefunction? Is it because of the size of the everyday object, so it won't behave so much like an electron? Theoretically, if the Planck constant is to be bigger, everyday object would start behaving more like particles in a quantum scale?</p>
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<p>There is a well known fact that a compact spacetime necessarily contains a closed timelike curve (CTC). Proof can be found in several books on GR (e.g. Hawking, Ellis, Proposition 6.4.2), and in essence goes like this:</p>
<p>The spacetime $M$ can be covered by open sets of the form $I^+(p)$, chronological future of the point $p \in M$ (note that <em>a priori</em> $p$ is not an element of the set $I^+(p)$, but such situation can happen in a presence of CTCs). Now, suppose that $M$ is compact. Then there is a finite subcover, say</p>
<p>$$\{ I^+(p_1), \dots, I^+(p_n) \}$$</p>
<p>The point $p_1$ is contained in $I^+(p_{k_1})$ for some $1 \le k_1 \le n$, the point $p_{k_1}$ is contained in $I^+(p_{k_2})$, and so on. Since this subcover is finite, eventually some point $p_{k_r}$ must belong to $I^+(p_{k_s})$, with $s \le r$. Then there is a future directed timelike curve going from $p_{k_r}$ to $p_{k_s}$ (since $s \le r$) and then from $p_{k_s}$ back to $p_{k_r}$ (since $p_{k_r} \in I^+(p_{k_s})$), which gives a closed timelike curve through $p_{k_r}$ (and $p_{k_s}$) in $M$. Q.E.D.</p>
<hr>
<p>The question that bothers me is: what is implicitly assumed about the spacetime $M$, by saying that the family of sets of the form $I^+(p)$ is indeed a covering of $M$?</p>
<p>Take for example a flat spacetime with compact space part and finite time direction, e.g. $M = [0,1] \times T^3$, where $T^3$ is a (spacelike) 3-torus. This is a compact manifold (since it is a product of two compact manifolds) and there are no CTCs. The loophole in the argument above seems to be in the fact that the "initial points", $\{0\} \times T^3$, are not covered by any set of the form $I^+(p)$.</p>
<p>One can easily modify this example by contracting the initial and final spacelike slices to points ("Big Bang" and "Big Crunch"); the resulting spacetime is still compact and contains no CTCs.</p>
<p>Do these manifolds fail to be "regular spacetimes" for some reason?</p>
| 9,448 |
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<p>I am aware of the differential form language proof of the fact that for arbitrary gauge transformations the Chern-Simons' term shifts by a WZW term (on the boundary). </p>
<p>But I am getting confused if in local coordinates I try to prove that under infinitesimal gauge transformations the Chern-Simons' term is invariant (may be upto total derivatives) </p>
<p>So I took my infinitesimal gauge transformation as $\delta A^a_\mu = \partial _\mu \epsilon ^a + f^{abc}A_\mu^b \epsilon ^c$ where my structure constants $f^{abc}$ are totally antisymmetric and cyclic. Then I look at the two terms that I have in the CS form, $\epsilon^{\mu \nu \lambda}A^a_\mu \partial_\nu A^a_\lambda$ and $\epsilon^{\mu \nu \lambda}f^{abc}A^a_\mu A^b_\nu A^c_\lambda$ and I vary them infinitesimally to get, </p>
<p>$\delta (\epsilon^{\mu \nu \lambda}A^a_\mu \partial_\nu A^a_\lambda ) = 2\epsilon^{\mu \nu \lambda}( \partial_\mu \epsilon ^a \partial _\nu A^a_\lambda + f^{abc}A_\mu ^b \partial _\nu A_\lambda ^a \epsilon ^c)$</p>
<p>$\delta (\epsilon^{\mu \nu \lambda}f^{abc}A^a_\mu A^b_\nu A^c_\lambda) = 3\epsilon^{\mu \nu \lambda}f^{abc} ( \partial _\mu \epsilon^a A_\nu ^b A_\lambda ^c + f^{adf} A_\mu ^d A_\nu ^b A_\lambda ^c \epsilon ^f) $ </p>
<ul>
<li>Now I can't see how any linear combination of the above two terms can make their combined variation 0 (or a total derivative). I would like to know as to what am I missing here. </li>
</ul>
| 9,449 |
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<p>In their novel <a href="https://en.wikipedia.org/wiki/Prisoners_of_Power" rel="nofollow">Inhabited Island</a> by brothers Strugatsky, the inhabitants of a planet think they live on the inner surface of a giant sphere because the refraction of the atmosphere makes the surface appearing concave. </p>
<p>At the same time, the atmosphere is suitable for a human to breathe.</p>
<p>I wonder whether it is actually possible?</p>
| 9,450 |
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<p>How does the size of humans compare to the size of other objects in the universe? Are we among the relatively large or the relatively small things?</p>
<p>My very preliminary research suggests that the smallest thing in the physical universe is the quantum foam which should be about Planck length or 10<sup>-35</sup> m. While the universe is about 10<sup>26</sup> m. This would seem to bias us toward the big side by about 9 orders of magnitude. </p>
<p>I am skeptical of this estimate because it is pretty close to the middle of the scale and I am always skeptical of measures that put humans in the center. So is there another way of thinking about/ quantifying this question?</p>
<p><strong>Edit:</strong> it seems that the smallest thing we have actual evidence of is a neutrino which is 10<sup>-24</sup> m, which would put us even closer to the center!</p>
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<p>I have a super stupid question about deriving Eq. (1.2.17) in Polchinski's string theory, vol 1.</p>
<p>The book seems to derive from </p>
<p>$$\tag{1.2.16} h_{ab}=\frac{1}{2} \gamma_{ab} \gamma^{cd} h_{cd} $$</p>
<p>to </p>
<p>$$\tag{1.2.17} h_{ab} (-h)^{-1/2} = \gamma_{ab} (-\gamma)^{-1/2}.$$</p>
<p>I cannot figure out how to manipulate the determinants $\gamma$ and $h$. Excuse me, would you provide at least any hint for the derivation?</p>
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<p>If an enormous asteroid, approximately the size of our moon (~2000-mile diameter), was passing close to Earth, how early might we detect something that large? Or alternatively, how close could something that large get without detection?</p>
<p>[edited to change term from meteor to asteroid]</p>
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<p>Apologies in advance if my question seems misinformed. I am a software developer, and neither quantum mechanics nor physics are my specialties.</p>
<p>From <a href="http://physics.about.com/od/quantumphysics/f/quantumcomp.htm" rel="nofollow">http://physics.about.com/od/quantumphysics/f/quantumcomp.htm</a>:</p>
<blockquote>
<p>The exact physical mechanism at work within the quantum computer is
somewhat theoretically complex and intuitively disturbing. Generally,
it is explained in terms of the multi-world interpretation of quantum
physics, wherein the computer performs calculations not only in our
universe but also in other universes simultaneously, while the various
qubits are in a state of quantum decoherence.</p>
</blockquote>
<p>And from <a href="http://www.cs.man.ac.uk/~toby/writing/PCW/q2.htm" rel="nofollow">http://www.cs.man.ac.uk/~toby/writing/PCW/q2.htm</a>:</p>
<blockquote>
<p>The amazing power of the quantum computer stems from the fact that if
you have a collection of qubits – a register – in which each qubit is
in an indeterminate state, then the register effectively represents
all possible numbers at once. If you then perform a single computation
on the register, the computation works on every possible number,
simultaneously. David Deutsch explains the process using the idea of
parallel universes – although we see only the single register in our
universe, it actually exists in many other universes too, one for each
of its possible states. By operating on the register in our universe,
we kick off computations in all the other unseen universes, and then
magically retrieve the answer.</p>
</blockquote>
<p>Assuming the multiverse interpretation of quantum mechanics is correct, does this mean that a quantum computer is capable of considering all possible solutions to a problem simultaneously because it delegates the consideration of (calculations for) each possible solution to a different universe?</p>
| 9,454 |
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<p>Why do people say that a <a href="http://en.wikipedia.org/wiki/Perpetual_motion" rel="nofollow">perpetual machine</a> is one that runs for eternity? Wouldn't a machine that runs taking no energy from outside but sometimes needs to be restarted, such as pendulum clock, be considered a perpetual machine? It just needs to be restarted one or two times a month. Given that physicists love approximations, wouldn't restarting a machine two times a month be higher order smaller term that we could throw away and call the pendulum clock a perpetual machine?</p>
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] |
<p>Why is it convenient to choose units kT=1? It came up in this paper: <a href="http://jfi.uchicago.edu/~leop/SciencePapers/scaling%20and%20universality%20in%20statistical%20physics.pdf" rel="nofollow">http://jfi.uchicago.edu/~leop/SciencePapers/scaling%20and%20universality%20in%20statistical%20physics.pdf</a> in page 4, equation (2.1). </p>
| 9,456 |
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] |
<p>Do low energy photons have greater entropy than high energy photons, or is the entropy of all photons the same (and the relationship between high and low entropy radiation is just a function of the number of photons - i.e. a closed system with more photons has higher entropy than the same sized system with fewer photons, regardless of the individual energies of the photons)? Is the entropy related to the number of different energies of a group of photons (i.e. do ten photons with the same wavelength have lower entropy than ten photons with different wavelengths)?</p>
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<p>I was under the impression that glass was transparent to light with a wavelength above ultraviolet, but when playing with my latest toy, an infrared thermometer which brought up a question <a href="http://physics.stackexchange.com/questions/121299/why-does-my-infrared-thermometer-say-the-sky-is-at-2-%C2%B0c">I asked here before</a>, expecting to be able to measure temperature via a mirror, I found that it would essentially just give me room temperature (or mirror temperature), at most slightly more, while via a piece of shiny metal I was able to measure the temperature of the object of interest (at most slightly less). </p>
<p>If it were the case that either most infrared radiation were transmitted or reflected I think I would be able to measure the temperature of the object, so I suspect that most heat radiation was absorbed. </p>
<p>Why does that happen while it doesn't happen for visible light? I think that for high frequencies electronic energy levels are available to absorb light, and that their unavailability for light in the visible range made glass transparent.</p>
<p>Why is it that for infrared energy levels are available (mechanical?) that apparently are not available for higher energy visible light?</p>
| 9,458 |
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<p>Several papers such as <a href="http://mail.sci.ccny.cuny.edu/~jtu/TuPRL03.pdf" rel="nofollow">this (warning, PDF)</a> and <a href="http://www.wiley-vch.de/vch/journals/2231/pss40/167_177.pdf" rel="nofollow">this (PDF again)</a> talk about how, near the electronic percolation transition for a metallic 2D film, the real part of the dielectric constant diverges (which, if you read those papers, gives rise to some neat effects).</p>
<p>I'm having trouble understanding how, though, aside from just the math. The 2nd paper mentions a more qualitative interpretation:</p>
<blockquote>
<p>Near the percolation threshold the metallic clusters are separated by thin dielectric regions. Each pair of nearest clusters forms a condenser [i.e., capacitor] whose effective surface tends to infinity near the percolation threshold. Then the effective capacity of the system diverges, too.</p>
</blockquote>
<p>Ok, let's assume for simplicity the "dielectric" here is vacuum, and use the form of a parallel plate capacitor, $C=A\epsilon_0/d$. By "effective surface tends to infinity", I'm guessing they mean that $d$ is going to zero (because the distance between clusters is going to 0), and thus $C$ is increasing to infinity.</p>
<p>From there, I'm a little confused. My best guess is that the impedance goes as $Z\propto 1/(i\omega C)$, and conductivity $\sigma \propto 1/Z$, and dielectric constant $\epsilon \propto i\omega \sigma$, so when the C increases, Z decreases, the imaginary part of $\sigma$ increases, and thus the real part of $\epsilon$ increases.</p>
<p>But I'm not really sure about that, and I'm having trouble reconciling it with what must happen after the percolation transition.</p>
<p>Does anyone have a better physical understanding of this?</p>
<p>thank you!</p>
| 9,459 |
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<p>I wanted to know how can I estimate the volume of the moduli space of a Riemann surface of genus $g$ and having $n$ marked points. I am reading some old string theory papers which discuss divergences in string perturbation theory. In this respect authors such as Gross, Periwal and Shenker say that the partition function in closed bosonic string theories asymptotically grows as $(2g)!$, where $g$ is the genus and they claim that this estimate can be obtained from the volume of moduli space. It would be great if anyone of you can suggest some reference where I learn about finding out the volume of moduli space of genus-$g$ surfaces with $n$ punctures. </p>
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<p>I've only recently started learning basic electrodynamics, but I don't understand why a loop of coil with a small area and a magnet falling through will produce a larger emf than a loop of coil with a larger area with the same magnetic field falling through. </p>
<p>To clarify, lets say you have a loop of coil and you drop a magnet through the coil. This is will produce an emf according to Faraday's law. Now if you have a loop of coil with a larger radius, and you drop the same magnet through at the same speed, it will produce a <em>smaller</em> emf. Why is this?</p>
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] |
<p>For a flow over a smooth sphere, does the wake region increases with increase in <a href="http://en.wikipedia.org/wiki/Reynolds_number" rel="nofollow">Reynolds number</a>? And if so, then why? </p>
| 9,462 |
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<p>Wave theory does not account for the photon model, which was developed only to explain quantum effects like photoelectric effect.
Then why do we talk about a photon's reflection and rarefaction, as that would require it to have wave properties? This has been mentioned here: (<a href="http://en.m.wikipedia.org/wiki/Photon" rel="nofollow">http://en.m.wikipedia.org/wiki/Photon</a>)
It is light that has wave-particle duality, not photon; it simply is a means of explaining the particle nature of light.</p>
| 309 |
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<p>For a potential $V$, how do we define the mass of a small fluctuation around its vacuum? For example I have the potential</p>
<p>$$ V_\mathrm{eff}(\phi) = \frac{1}{2} \left(\frac{\rho}{M^2} - \mu^2\right) \phi^2 + \frac{1}{4} \lambda \phi^4. $$</p>
<p>What is the definition of the mass of small fluctuations around the symmetry-breaking vacuum and what is the physical meaning of it?</p>
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<p>Say there is a horizontal rod, where a force $P$ is applied to the right hand side of the rod to drag it horizontally to the right. If I drew a force diagram, I'd put some vector with a length of $\| \mathbf{P} \|$ coming out of the right end of the rod. In actuality, if the rod is composed of ten particles, then is $\| \mathbf{P} \|$ distributed evenly? So if I felt like creating a more accurate force diagram, then each particle would have a smaller vector going to the right with magnitude $\| \mathbf{P} \|/10$?</p>
<p>Is this true? Is the reason we usually only put 1 vector of length $\| \mathbf{P} \|$ for simplicity? Likewise when we assume the force of gravity $F_g$ on an extended object can be put at the center of gravity of that object as if it was a point mass. But in actuality, there are like a billion really small $F_g$ vectors on each particle?</p>
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<p>I want to find out how many calories are burned on any particular hike by any particular by using a formula. I figure it's probably a simple physics question, but since I never took physics, I don't know the math to do it. </p>
<p>My premise is that you will expend (A) energy (calories) when moving (B) weight (C) distance on a level path. And also (X) energy (calories) when moving (Y) weight (Z) distance when moving up against gravity. </p>
<p>The formulas I have found will add (A)+(X) to get the totals calories burned. Trouble is, the results vary greatly, and I have no way of judging which one is right.</p>
<p>Now, I know there lots of variables, but for this exercise, we'll skip the small ones. Here's what I'm thinking:</p>
<ol>
<li>Obviously there's resistance. At first I was dismissing resistance as being too minor when only traveling 2-4 MPH, but then I realized that without resistance, we'd just keep moving. Now, resistance will be based on the front profile size. Can we simplify things by saying X weight = Y profile?</li>
<li>Assuming X weight = Y profile. A backpack would add weight, but not added profile. So we'll say no added weight for the formula.</li>
<li>Up & down motion for each step. I figure that makes the formula a lot more complex because we have to deal with the height & stride of each person. So I guess we'll skip that.</li>
<li>A fit person burns less calories (more efficient muscles) than an out of shape person doing the same hike. So we'll pick a good average. </li>
<li>Downhill is considered the same exercise as level walking.</li>
<li>Age & sex may be a factor, but I don't have those numbers, so can be ignored for now. I may later ask on a health forum for those factors.</li>
<li>There's probably others I haven't thought about.</li>
</ol>
<p>There are many sites on the internet that calculate this for you, but there's 2 problems with them.</p>
<ol>
<li>The majority want to ask you your hiking speed and time hiked. But
most people don't their hiking speed, and some sites are vague with
"brisk walk". Then all they do is convert speed with time to get
distance.</li>
<li>Even the ones that factor weight, distance, & elevation gain, I get widely different answers.</li>
</ol>
<p>So, what I would like to get, if someone is in the mood to figure it out, is a formula for calculating calories burned. </p>
<p>Thanks</p>
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<p>Is it possible to use a set of cylindrical mirrors and software correction to build an optical reflector telescope, instead of using a paraboloid reflector? By 'cylindrical' I actually mean a planar mirror curved to a parabolic profile.</p>
<p>In other words, rather than focusing light in 360 degrees in one step, is it possible to focus it in two steps -- first along one axis and then using a perpendicular 'cylindrical' mirror to focus it along the other axis?</p>
<p>Does this introduce optical distortions, and if so, can software correction be used?</p>
<p>The reasoning is that 'cylindrical' reflectors are far easier and cheaper to fabricate (either using sheet metal techniques or linear grinding methods) than traditional ones. They are also easier to stack together, launch into orbit and deploy once it reaches stable orbit (for space telescopes).</p>
<p>A Google search revealed only a single document with an attempt to describe such a setup, but it is over a decade old: <a href="http://www.bruno.postle.net/2001/telescope/">http://www.bruno.postle.net/2001/telescope/</a></p>
<p><img src="http://i.stack.imgur.com/dc5DA.png" alt="enter image description here"></p>
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<p>In my understanding, there exists a critical mass for which a star needs to be in order for it to collapse into a black hole. This also applies to a certain critical density of gas in order for stars to form in the first place. However, given a massive star or a massive amount of gas, it must have a relatively large gravitational field and therefore will interact with dark matter, attracting dark matter towards itself. This makes me question, doesn't the critical mass or critical density that we've figured out before our knowledge of dark matter now need to take that into account?
OR
Shouldn't the critical mass and critical density resemble the following pseudo equations?</p>
<p>Critical mass (normal and dark) for a star to turn to black hole = A*normal matter mass + B*dark matter mass, where A and B are fractional amounts</p>
<p>Critical density for a cloud of gas (normal and dark) for star formation = C*normal matter mass + D*dark matter mass, where C and D are fractional amounts</p>
<p>So if the answer is yes, then what are the fractions, A, B, C, and D? Moreover, What kind of experiment or calculation would allow us to figure out these fractions?</p>
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<p>We owe a debt to Bell labs with Shockley, Bardeen and Brattain for inventing the semiconductor.
All have now passed on. I remember seeing my first transistor radio in the early 60's.
Shockley developed the very important diode equation.
I was reading through one of my good old engineering books where the dynamic resistance of a diode is derived. I'm trying to understand this because it's basic to many applications for transistors, Current mirrors, Long tailed pairs etc.
The dyamic resistance is used to get the volt drop over the small signal ac resistance.
They give an equation :</p>
<p>1) $I =Is(e^{eV/kT}-1)$</p>
<p>I is the diode current</p>
<p>V is the diode voltage</p>
<p>K is Boltzmans contstant</p>
<p>T is the room temp in Kelvin</p>
<p>Is is the reverse leakage current</p>
<p>They then derive the dynamic resistance (as dV/dI) and get an answer as roughly 1/40I.</p>
<p>I can't get the same answer that they do.
Instead,I have to use this equation from Wikipedia to get the same answer.</p>
<p>2) $I=Is(e^{V/Vt}-1)$</p>
<p>V is the diode voltage</p>
<p>Vt is the thermal volts. 25mv at room temperature.</p>
<p>Are they using the wrong equation in (1) , or have i worked it out wrong?</p>
<p>Thanks in advance.</p>
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<p>Potential flow around a circular cylinder is a classic solution. But I am wondering if there is any solution similar to this for the flow past a square cylinder?</p>
| 9,469 |
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<p>and another exam question, this is about current research:</p>
<blockquote>
<p>Interference of matter waves has been studied using ultra-cold atoms. The phase of a matter wave for free-falling cold-atoms at time $t$ and height $y$ is given as
$$\phi(y,t) = \phi(y_0,t) + Et - \int_{y_0}^y dy' k(y')$$
where $E$ is the total energy of individual coherent atoms and $\hbar k(y)$ is the semi-classical momentum of falling atoms at height $y$. A recent experiment observed coherent matter waves continuously emitted from two vertical micro-traps (you can neglect the size of the traps) with height $-\lambda/2$ and $\lambda/2$. The atoms are emitted at rest from the traps.</p>
<p>A) For a given height $y$ below the two micro-traps, derive the condition of constructive and destructive interference.</p>
</blockquote>
<p>Well, from basic wave mechanics I know that the phase difference of the two waves has to be an even (constructive) or odd (destructive) multiple of $\pi$, but apart from that I find the question poorly worded: What does $t$ refer to? Total time in laboratory frame, or time since leaving the trap? And what about the function $\phi(y_0,t)$? If I don't know its form, can I say anything meaningful about the interference pattern at all?</p>
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<p>I am reading <em>A Brief History of Time</em> by Stephen Hawking, and in it he mentions that without compensating for relativity, GPS devices would be out by miles. Why is this? (I am not sure which relativity he means as I am several chapters ahead now and the question just came to me.)</p>
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] |
<p>A superconducting wire has no electrical resistance and as such it does not heat up when current passes through it. Non-superconducting wires can be damaged by too much current, because they get too hot and melt. But since current is the flow of electrons, surely unlimited electrons can't flow?</p>
| 9,472 |
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<p>I have two parallel non conducting charged planes with opposite charges $6\mu C/m^2$, area $A = 3m^2$ and distance between the planes $d = 0.004 m$. I know the potential between these two planes is $$V = \frac{\sigma\times d}{\epsilon_0}$$
But if I put a conducting sheet of thickness $h = 0.001 m$ with the same area $A = 3m^2$ between these two planes, the new potential between the the planes would be</p>
<p>$$V = (2\times\frac{\sigma}{2\epsilon_0} + 2\times\frac{2\sigma}{2\epsilon_0})\times d$$</p>
<p>Is this right? or Am I wrong somewhere? I mounted this schema.</p>
<p><img src="http://i.stack.imgur.com/j1peM.jpg" alt=""></p>
<p>Is that right? </p>
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<p>The geometry of my semiconductor device is given below. The blue regions are gold, the grey ones - gallium arsenide (n-doped to $2.9 \times 10^{15} \mathrm{cm^{-3}}$). The dimensions are μm, i.e. it is 500μm wide, the bottom GaAs is 400μm thick, then a layer of gold follows, then a 15μm layer of GaAs, and a 5μm layer of Au on the top:
<img src="http://i.stack.imgur.com/4zaKq.png" alt="geometry"></p>
<p>The thinner strip of GaAs is a laser medium where radiation of 3THz is emitted. I would like to investigate the effect of the thickness of the thin layer of gold on the optical mode profile. I simulated the profile for two thicknesses: 20nm and 200nm. Below are the plots of the normalised E field, first 2D and then a cross section along x=0 line. Arc length is simply the y axis of the first plot.</p>
<p><strong>20nm</strong>
<img src="http://i.stack.imgur.com/jsZIT.png" alt="20nm 2d">
<img src="http://i.stack.imgur.com/3JqiX.png" alt="20nm 1d"></p>
<p><strong>200nm</strong>
<img src="http://i.stack.imgur.com/Oe2cF.png" alt="200nm 2d">
<img src="http://i.stack.imgur.com/XqQJh.png" alt="200nm 1d"></p>
<p>In the first (20nm) case, |E| decreases from 450V/m to 90V/m (5x). In the second (200nm) case, |E| decreases from 2800V/m to 10V/m (280x).</p>
<p>The properties of the materials are as follows (Drude model for GaAs, literature for Au):</p>
<pre><code>+----------+-------+--------+--------------+
| Material | Im(ε) | Re(ε) | conductivity |
+----------+-------+--------+--------------+
| Au | 0 | -51746 | 4.1e7 S/m |
| GaAs | -0.17 | 12.98 | 4 S/m |
+----------+-------+--------+--------------+
</code></pre>
<p>The field should decrease as $\exp(-\alpha z)$ (Beer's law), where $\frac{1}{\alpha} = \delta \approx 50\mathrm{nm}$ is the skin depth for 3THz in gold - this is calculated from:
$$\alpha = \frac{1}{\delta} = \frac{\kappa \omega}{c}$$
where $\kappa$ is the known value for $Im(\tilde{n}) = 319$ (Ordal et al., <a href="http://dx.doi.org/10.1364/AO.26.000744" rel="nofollow">http://dx.doi.org/10.1364/AO.26.000744</a>). Hence:
$$E_2(20nm) = 450 \exp(-20/50) \approx 258$$
$$E_2(200nm) = 2800 \exp(-200/50) \approx 11$$</p>
<p>The second result looks credible, given the uncertainty $\kappa$. However, the first one is 3x higher than I see in the simulation.</p>
<p>To check this, I simulated the system for different thicknesses of the gold layer, and fitted the data to the equation:
$$\frac{|E_2|}{|E_1|} = A \exp \left(-\frac{z}{\delta} \right)$$</p>
<p>I got the following plot:
<img src="http://i.stack.imgur.com/YQzZJ.png" alt="fit"></p>
<p>And the fit yields these parameters:
$$\frac{|E_2|}{|E_1|} = 0.32 \exp \left(-\frac{z}{41\mathrm{nm}} \right)$$</p>
<p>So $\delta=41\mathrm{nm}$ is not too far from the calculated value. What I am trying to understand now is:</p>
<p><strong>Where does the factor of 0.32 come from?</strong> It obviously matters much more for lower thickness of gold, which I am interested in. Does it have something to do with the normalisation? Boundary conditions?</p>
<p>I have realised that I should be looking at $\mathrm{E_y}$ rather than $\mathrm{|E|}$, but I checked it and the values look the same.</p>
<p>COMSOL FEM software was used to visualise the problem below. The question is related - albeit different - to this one: <a href="http://physics.stackexchange.com/questions/107282/the-skin-effect-and-the-reflectivity-of-gold">The skin effect and the reflectivity of gold</a></p>
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<p>$$\mathbf D = \varepsilon \mathbf E$$
I don't understand the difference between $\mathbf D$ and $\mathbf E$.
When I have a plate capacitor different medium inside will change $\mathbf D$, right?
$\mathbf E$ is only dependent from the charges right?</p>
| 9,475 |
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<p>I don't really know much about Quantum mechanics, but would like to know one simple fact.</p>
<p>The state function $\Psi(r, t)|$ whose magnitude gives the probability density of the position of the particle and the magnitude of its ($\Psi(r, t)$) fourier transform gives probability density of its momentum. Is there any rule that these state functions are smooth (possess infinite order derivatives everywhere) (derivatives of all orders exist)?</p>
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<p>Are there any objects out there that have since vanished, but because of their distance and the travel time for light, can be seen <em>with the naked eye</em>? Or are those things only visible with magnification? How much magnification at least?</p>
| 9,477 |
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<p>While an object is moving, relativity will say it weighs more, especially so as it approaches light speed. </p>
<p>The increase in energy is then easily sensed as an increase in mass
(Almost as a rock in your hand feels 'lighter' than the rock that falls upon your hand, before the energy of velocity is transferred and potential difference becomes zero.)</p>
<p>Is this increase in 'mass' measurable by our 'balance type' weight scale, or is it better off being considered an 'effect' of the potential difference between reference frames in which the mass appears momentarily heavier in relation to an equal mass with 'zero' velocity? Very curious. **Almost like 'electricity' where the measured ground connection is considered 'zero' voltage, although we know the ground has a charge by evidence of a magnetic field and telluric currents, we still consider this electrical state as a 'relative' 'zero', right?
Same thing as momentum, at least as I currently perceive it; pure potential difference and totally relative, whether it's 30-90 volts or -30-30 volts, its the same 60v difference.</p>
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<p>I am in need of a book (preferably) or other reference material which would help technicians determine whether porosity exists in the fracture face of a tensile test bar for a variety of different types of metal. The reference would not necessarily need to show fracture faces, just porosity in general. Types of metals would need to include stainless steels and nickel and iron based superalloys. </p>
<p>To state this in the form of a question: I have some metal, how do I know if it contains porosity?</p>
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<p>I'm looking for some good explanation on leading logarithmic approximation (LLA) in QCD; in particular, what types of contributions can be neglected while assuming LLA?</p>
| 9,480 |
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<p>I have been thinking about this question for quiet a long time but, couldn't make up an answer myself. Usually when someone asks about why a system is the way it is we answer it by saying that it is the most stable lowest energy state(to know about why a system should be at its lowest energy state for its stability click <a href="http://physics.stackexchange.com/questions/87371/why-a-system-should-be-at-its-lowest-energy-state-for-its-stability">here</a>). But, what about the uncertainty principle? What benefit does the universe/nature get by following this principle? </p>
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<p>If a particle is being expelled from the Black Hole and an antiparticle is being driven into it, shouldn't the opposite occur as well and in the same frequencies?</p>
<p>I mean Black holes should emit antiparticle radiation as well and gather energy from it and the black hole energy shouldn't change, right?</p>
<p>What am I missing?</p>
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<p>From my bookish knowledge all planet revolve around the sun in anticlockwise direction and electron also moves in the anti clock around the nucleus. Is there any specific reason behind the stability of anticlock wise direction? </p>
| 9,483 |
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<p>In this problem there is a cylinder that is closed at both ends and has thermally insulated walls. It is divided into 2 parts by a movable, frictionless, thermally insulated piston. There is a heating coil in the left side of the cylinder and there are $N$ molecules of an ideal gas in each compartment. The original temperature and volume on each side are To and $V_o$. The left side is slowly warmed with the coil until its pressure doubles.</p>
<p>How much work was done on the gas on the right?
I calculated the final temperature and volume of each side after the left side is heated but I'm unsure about the work calculation. Would the work simply be $W = -P(V_f - V_o)$ where $P$ is 2 times the original pressure (so its $2NkT_o/V_o$) or do I have to do something with the pressure since it changes from $P$ to $2P$?</p>
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<p>I have been playing around with Tesla coils for a while now, but one limiting factor for portability is the fact that it requires an earth connection on the secondary coil.</p>
<p>Since a Tesla coil works with resonant inductive coupling, I was wondering if it would be theoretically possible to wirelessly couple the secondary coil to the earth by letting it resonate with one of the Schumann resonances present in the earth, achieving the same result as you would when you connect it via a wire to the earth.</p>
<p>Thanks!</p>
| 9,485 |
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<p>I don't want this to be a "do my home work question" so please tell me how I can make this question helpful for other people.</p>
<p>In my physics assignment I found the question below. I'd think that both FBD's and $F_{net}$s are the same for each scenario; since the track is friction-less the rider doesn't need to input anything to maintain a constant velocity. </p>
<p>If it helps, the answers for 'c' were found to be: $ac = g\tan(\theta)$, $a = g\sin(\theta)$ for each respective scenario.</p>
<p>Any help is much appreciated.</p>
<p><img src="http://i.stack.imgur.com/7A93J.png" alt="Problem text"></p>
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<p>I understand from my very limited knowledge of relativity that an object traveling at relativistic speeds essentially experiences the progression of time slow to a crawl. Since, according to <a href="http://en.wikipedia.org/wiki/General_relativity" rel="nofollow">Einstein's GR theory</a>, gravity bends spacetime, does that mean that time is also slowed down by large amounts of gravity?</p>
<p>If I were in a place where there was absolutely no gravity in any way shape or form, would I experience time passing more quickly, and if so, by how much? </p>
| 310 |
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<p>I was wondering when I boil water in a pot it only shakes too much while boiling. But I could not figure out why tea rises in the pot when we boil it. it is also a liquid but it starts rising up till it jumps out of the pot and tries to kill the flame. </p>
<p>I think this happens due to change in the adhesive properties of the water due to addition of milk, sugar and tea leaves to it. Am I right? if not what the reason behind it?</p>
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<p>"Now we turn our attention to what happens in $S'$, in which the particle is at rest and the wire is running past (toward the left in the figure) with the speed $v$. The positive charges moving with the wire will make some magnetic field $\ B'$ at the particle. But the particle is now at $rest$, so there is no magnetic force on it! If there is no magnetic force on the particle, it must come from an electric field. It must be that the moving wire has produced an electric field. But it can do that only if it appears $charged$-it must be that a neutral wire with a current appears to be charged when set in motion."</p>
<p>So i got this from Feynman's lectures in physics volume 2 pg 13-7 and 13-8. I find it hard to believe that the particle experiences no magnetic force due to the moving wire although it's stationary. Since this is classical and not quantum electrodynamics if we assume the charged particle to have inertia, then by Newton's second law, the particle will remain at rest until acted upon by an external force, which i assume is the $magnetic$ $wind$ created by the moving wire past the stationary charged particle. Could someone please clarify??</p>
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<p>Quantum shot noise (either optical intensity noise or electrical current noise) described by the noise spectral density of $2 e I$ (electrically) or $2 h \nu P$ (optically).</p>
<p>So it is white noise. I know this basically comes from the derivation, where we model electron or photon events as infinitely short (with delta impulses). See e.g. Eq 1 in <a href="http://123.physics.ucdavis.edu/shot_files/ShotNoise.pdf" rel="nofollow">http://123.physics.ucdavis.edu/shot_files/ShotNoise.pdf</a>.</p>
<p>$$I(t)=\sum_j q \;\delta(t-t_j)$$ </p>
<p>So far so good, but can this really be true in an exact sense? A consequence would be that with growing bandwidth, the fluctuation (variance) grows indefinitely. That cant be physically correct.</p>
<p>So things I would like to know specifically:</p>
<ul>
<li><p>Are there any experimental results where the spectral noise density of shot noise is measured? If e.g. there is a (nonzero) transit time of electrons, we should see some interesting things happen in the noise density at the inverse transit time. has something like this been observed? </p></li>
<li><p>Is there a "more correct" (quantum physical) theory to derive shot noise? I.e. gives the correct spectral density? If yes, what is it and what is the idea behind the theory?</p></li>
<li><p>Might there be a fundamental reason why it is impossible to measure shot noise at high frequencies? e.g. there is a fundamental limit for the gain bandwidth product of amplifiers (with which we could observe the shot noise at arbitrary frequencies in, for example, a diode)?</p></li>
</ul>
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<p>My old theoretical physics professor used to say that (already) (special?) relativity shows that "the future is not predictable". Any ideas how this should be interpreted?</p>
| 9,490 |
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<p>A wet object or a volume of water will decrease in temperature due to the effect of evaporation. We understand this to be because of the molecular kinetics, where the faster water molecules escape and become a part of the air. Since those that escape have more energy than the average, the process decreases the temperature of the water, but how does it affect the temperature of the air?</p>
<p>I can imagine two different factors:</p>
<ol>
<li>The escaped water molecules have a higher energy than the air molecules, since the air was previously at equilibrium with the water</li>
<li>The binding energy that makes water liquid and gives it surface tension reduces the energy at which a molecule escapes</li>
</ol>
<p>While #1 would seem to predict that the air becomes hotter at the expense of the water becoming colder, #2 should imply that the net thermal energy of the entire system decreases because some of the kinetic energy is converted to a form of potential energy (not sure to call it surface tension, chemical potential, or something else). These two factors are in conflict, so the answer is non-trivial to me.</p>
<p>Let's not consider long term factors like convection eventually cooling the air. Considering only the action of a molecule escaping the surface of water through evaporation in a closed system, how does that affect the temperature of the gas it escapes into?</p>
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<p>A solid can exist in two phases, with energies</p>
<p>$$U_{1}(S,V)=\frac{S^2}{a_1}+b_{1}V(V-2V_{0})$$</p>
<p>$$U_{2}(S,V)=\frac{S^2}{a_2}+b_{2}V(V-2V_{0})$$</p>
<p>where $a_{1},a_{2},b_{1},b_{2},V_{0}$ are positive constants, with $a_{1}>a_{2}$ and $b_{1}<b_2$.</p>
<p>Consider a phase transition between the two phases at pressure $P = 0$. </p>
<p>At what temperature, $T_0$, does this occur?</p>
<p>I calculated the enthalpy, Helmholtz free energy, and Gibbs free energy for each side, I'm just having trouble figuring out what I need to do with them in order to calculate the temperature a phase transition would occur. Does someone know what I need to do with those values in order to get $T_{0}$?</p>
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<p>Why can't the work done by a <a href="http://en.wikipedia.org/wiki/Conservative_force" rel="nofollow">non-conservative force</a> be zero? The displacement along a closed path is always zero. So, whatever be the type of force, variable or constant, the work has to be zero. Why do we need to calculate the work done for individual paths?<img src="http://i.stack.imgur.com/jtrI2.jpg" alt="enter image description here"></p>
<p>This is a non-conservative force that starts from $A$ moves via Path 1 to $B$ and then back to $A$ via Path 2. Since the displacement is anyways going to be zero, why can't work done be zero?</p>
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<blockquote>
<p><em>The curvature of Mars is such that its surface drops a vertical
distance of 2.0 meters for every 3600 meters tangent to the surface.
In addition, the gravitational acceleration near its surface is 0.4
times that near the surface of the Earth. What is the speed a golf
ball would need to orbit Mars near the surface, ignoring the effects
of air resistance?</em></p>
</blockquote>
<p>In the Gre question, it can be solved by using the equation $$x=x_0+V_{x_0}t +\frac12 a_x t^2$$ and $$y=y_0+V_{y_0}t +\frac12 a_y t^2$$</p>
<p>Where $a_y= -g \times .4$, $x=3600$, $y=2$.</p>
<p>But I want to solve it by the using of the equation of catelite planet.
Which is $$v= \sqrt{\frac{g R^2}{R+h}}.$$
Is it possible to do this in the above equation. I am saying this because h is not given in the question. </p>
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<p>I'm reading the following <a href="http://onlinelibrary.wiley.com/doi/10.1111/j.2153-3490.1967.tb01462.x/pdf" rel="nofollow">paper</a> and do not entirely understand a point that the author is trying to make. In page 99 (2 of article) the author refers to the following equation:</p>
<p>$$ Q^{*} = \beta S^{*}e^{-\beta z} + 2B^{*} \delta \left(z\right) $$</p>
<p>and in the following paragraph he states that</p>
<p>"The first term on the right represents the convergence of the penetrating component S^{*} of solar radiation"</p>
<p>My immediate question is does "first term" refer to the $2B^{*}\delta\left(z\right)$ or does it refer to "2B"? i.e. does rhs mean rhs of the equal sign or what? </p>
<p>In addition, can anyone explain, in layman's terms, what this equation is referring to? </p>
| 9,495 |
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<p>What exactly is the relationship between the continuity equation and the wave equation?</p>
<p>Suppose $J^\mu$ is a contravariant vector that satisfies the <a href="http://en.wikipedia.org/wiki/Continuity_equation#Special_relativity" rel="nofollow">continuity equation</a> $\partial_\mu J^\mu=0$. Let $J^\mu$ be defined by $J^\mu=\partial^\mu\varphi$, where $\varphi$ is some <a href="http://en.wikipedia.org/wiki/Lorentz_scalar" rel="nofollow">Lorentz scalar</a>. Performing a simple substitution yields $\partial_\mu\partial^\mu\varphi=0$, or $\square\varphi=0$, where $\square$ is the <a href="http://en.wikipedia.org/wiki/D%27Alembert_operator" rel="nofollow">D'Alembert operator</a>. This seems to be a manifestation of the <a href="http://en.wikipedia.org/wiki/Wave_equation" rel="nofollow">wave equation</a>. Is this a correct derivation? If so, what is the physical interpretation behind it?</p>
| 9,496 |
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<p>Suppose you have a set of charges that are Newtonian (not quantum and not fast-moving) point particles. They are subject to known (but not necessarily constant) external forces ($F_{ext}$), as well as mutual electrostatic forces ($F_s$).</p>
<p>We want to estimate the <em>electromagnetic</em> forces ($F_m$) on each charge; these forces will over time sap energy form the system. We want an estimate of $F_m$ that is accurate, in other words the <em>relative</em> error in $F_m$ goes to zero in the Newtonian limit ($F_m$ itself will go to zero as well, thus my specification of relative and not absolute error). Is there a way to estimate this easily (without resorting to Maxwell's equations)? If so how?</p>
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<p>For example, let's consider a quadratic fermionic Hamiltonian on a 2D lattice with translation symmetry, and assume that the Fourier transformed Hamiltonian is described by a $2\times2$ Hermitian matrix $H(\mathbf{k})=a(\mathbf{k})\sigma_x+b(\mathbf{k})\sigma_y+c(\mathbf{k})\sigma_z $ and has a finite energy gap, then the Chern number $N$ can be determined.</p>
<p>If $H(-\mathbf{k})=H(\mathbf{k})$ holds for all $\mathbf{k}\in BZ$, then the Chern number $N$ is always an <strong>even number</strong>, am I right? This seems to be true from the geometrical interpretation of Chern number as a winding number covering a unit sphere, but I have not yet found a rigorous mathematical proof.</p>
<p><strong>Remark:</strong> The necessary condition <em>finite energy gap</em> ($\Leftrightarrow$ <em>The map $(a(\mathbf{k}),b(\mathbf{k}),c(\mathbf{k}))/\sqrt{a(\mathbf{k})^2+b(\mathbf{k})^2+c(\mathbf{k})^2}$ from BZ(2D torus) to the unit sphere is well defined</em>) is to ensure that the Chern number/winding number is well defined.</p>
| 9,498 |
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<p>I am writing a novel, and although I have a background in physics, I am unsure of the exact equations to use.</p>
<p>Specifically:</p>
<p><strong>For a photon-sail ship, how powerful would a driving laser need to be in order for the ship to reach an acceleration of ~0.5g?</strong></p>
<p>Assuming we are only talking within the solar system here, and assuming that the ship is heavy enough to carry passengers.</p>
<p>Would such a thing ever be feasible? I am trying to strike the right balance between fun and feasibility for some method of regular interplanetary transport (think the space equivalent of commercial jets).</p>
| 9,499 |
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<p>The following is defined, where $\epsilon \to 0^+$ is a cutoff:
$$ \mathcal{F}(Z)=\int_{\epsilon}^\infty \frac{\mathrm{d}s}{s}
\frac{1}{\sinh^2 s/2} e^{-sx}. $$</p>
<p>Question: how do we see that $\mathcal{F}$ has a
perturbative + non-perturbative expansion (in $x \gg 0$) given by</p>
<p>$$ \mathcal{F}(Z) = \sum_g \mathcal{F}_g x^{2-2g} + O(e^{-x}),$$
for some coefficients $\mathcal{F}_g$ ?</p>
<p>Reference: the issue is discussed in <a href="http://arXiv.org/abs/hep-th/9809187" rel="nofollow">arXiv:hep-th/9809187</a> eq. (3.2) and below. The book <em>Quantum Field Theory</em> by Itzykson and Zuber eq. (4-117) and below should help clarify.</p>
<p>It should be possible to see this kind of expansion by rescaling the variable $s$ in the integral and making a Taylor expansion of the $\sinh^2$ in the denominator. The non-perturbative terms should arise from picking up poles at imaginary values of $s= 2\pi i n.$</p>
<p>Note that the $\epsilon$ parameter is supposed to ensure convergence of the integral.</p>
<hr>
<p>I think I found a partial answer in the paper <a href="http://arxiv.org/abs/0907.4082" rel="nofollow">http://arxiv.org/abs/0907.4082</a> paragraphs 2.2 and 2.4, although I'm not 100% sure how one relates formula (2.20) here with (2.1) in <a href="http://arxiv.org/abs/hep-th/9812127" rel="nofollow">http://arxiv.org/abs/hep-th/9812127</a></p>
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<p>Does a phase transition occur as a gas is heated to create a plasma? If so, is this a first or second order phase transition?</p>
<p>Also, does the presence of a phase transition depend on the pressure or composition? It seems to me that in the dilute limit (i.e. low pressures), no phase transition should occur because the fraction of atoms that are ionized will follow a Boltzmann distribution, which is a smooth function of temperature. However, the presence of <a href="http://link.aps.org/doi/10.1103/PhysRevLett.71.3826" rel="nofollow">phase transitions in Debye-Hückel theory</a> seems to suggest that a gas-plasma transition could occur at higher pressures.</p>
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<p><strong>Context</strong></p>
<p>I have been going through <a href="http://arxiv.org/abs/1302.5843" rel="nofollow">Ising formulations of many NP problems</a> by Andrew Lucas. In section 9 on page 22, the author gave an Ising formulation of the graph isomorphism problem. Here I mention it briefly.</p>
<p>Given two graphs, $G_1$ and $G_2$ of same $N$ number of vertices, the Ising Hamiltonian is,</p>
<p>$$H = H_A + H_B$$. Here,</p>
<p>$$H_A = A \sum_v \left(1 - \sum_i x_{v, i}\right)^2 + A \sum_i \left(1 - \sum_v x_{v, i}\right)^2 $$</p>
<p>and </p>
<p>$$H_B = B \sum_{ij \not\in E_1} \sum_{u v \in E_2} x_{u, i} x_{v, j} + B \sum_{ij \in E_1} \sum_{u v \not\in E_2} x_{u, i} x_{v, j}$$</p>
<p>Here $x_{v,i}$ is a binary variable which is $1$ if vertex $v$ in $G_2$ gets mapped to vertex $i$ in $G_1$. Although not mentioned explicitly in that section of the paper, I assume $v$ is an element of the set of vertices $V$ of a graph $G$ and $E$ is the set of all edges. Moreover, $A$ and $B$ are the Ising interactions.</p>
<p><strong>My questions</strong></p>
<ol>
<li><strike>For $N$ vertices, $H_A$ will contain at least one term which consists of $N^2$ Ising variables. So, it will not be quadratic for any non-trivial graph. Say $N = 2$. Then at least one term will have $x_{1,1} \times x_{1,2} \times x_{2,1} \times x_{2,2}$. Is there a known way to decompose it into quadratic form so that it could be implemented on a hardware which only accommodates nearest neighbor interaction?</strike></li>
<li>As $x_{v,i}$ is a boolean variable, any term, after expanding $H_A$, consisting the form $x_{v,i}^2$ should be same as $x_{v,i}$. Am I getting it right?</li>
</ol>
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] |
<p>I want to know if a household AC current flowing through a simple copper wire will radiate EM waves? If yes, up to what range and can they be called radio waves? </p>
| 9,503 |
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<p>I am studing the Grover operator. Let be $x_0$ the marked element. Let be $t_f = \pi/2\omega$. The measurement of the register in the computational basis return $x_0$ with probability </p>
<p>$$p_{x_0}(t_f)=\langle x_0|U^{t_f}|D\rangle|=1-1/N$$</p>
<p>Where $|D\rangle$ is a diagonal state state, $U$ the grover operator, $\cos \omega = 1-2/N$ and $N=2^n$.</p>
<p>My lecture say $p_{x_0}(t_f)$ is a lower bound for $p_{x_0}(\lfloor t_f\rfloor).$ $\lfloor t_f\rfloor=\lfloor (\pi/4)\sqrt{N}\rfloor + O(1/\sqrt{N})$ was obtained taking the asymptotic expansion in $N$. My question is Why $p_{x_0}(t_f)$ is a lower bound for $p_{x_0}(\lfloor t_f\rfloor).$?</p>
| 9,504 |
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<p>I have always wondered and once I even got it, but then completely forgot. I understand that gravity causes high and low tides in oceans, but why does it occur on the other side of Earth?</p>
| 9,505 |
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<p>I am solving this problem from a previous year question paper</p>
<blockquote>
<p>A series RLC circuit has resistance of 1k at resonant frequency and half power frequencies are
20kHz and 100kHz. Determine (i) The bandwidth (ii) The value of L and C iii) Show the nature of
variation of impedance and power factor in a series resonant circuit.</p>
</blockquote>
<p>I solved the first two. For the third one I came to the conclusion that as long as impedance decreases power factor increases from the equation
$$\cos \phi=\cos(\arctan \frac {X_C-X_L}{R})$$
As the numerator decreases the value of $\arctan$ decreases hence power factor increases. Also power factor will decrease after the circuit passes the resonance condition. I do not know how to graph this. Can anyone provide me some hint on how I can approach the last problem?</p>
| 9,506 |
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<p>When the spin of a quantum entangled particle is measured, is it only possible to do an instantaneous measurement, or can a particles spin be held in a collapsed state by constantly observing it?</p>
<p>In other-words, can you inspect particle A's spin and continuously observe it while sampling B's spin multiple times?</p>
| 9,507 |
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<p>I'm trying to compute the expectation value of energy for a certain state in an <a href="http://en.wikipedia.org/wiki/Particle_in_a_box" rel="nofollow">infinite potential well</a> but I'm getting contradictory answers.</p>
<p>The well has potential
\begin{align}
V(x) = \left\{
\begin{array}{lr}
0 & : 0 < x < L\\
\infty & : \text{ elsewhere}
\end{array}
\right.
\end{align}
which has eigenstates $\phi_n(x) = \sqrt{\frac{2}{L}} \sin(\frac{n \pi x}{L})$ with corresponding energies $E_n = \frac{n^2 \pi^2 \hbar^2}{2mL^2}$.</p>
<p>Now, consider the state
\begin{align}
\psi(x) = \left\{
\begin{array}{lr}
0 & : 0 < x < L/2\\
\sqrt{\frac{2}{L}} & : L/2 \leq x < L\\
0 &: \text{ elsewhere}
\end{array}
\right.
\end{align}
We want to compute $\langle H \rangle$ for this state. One way to do this is simply using the definition: $\langle H \rangle = \int_{0}^L \psi^*(x) H \psi(x) dx$. The problem with this though is that $\psi(x)$ is piecewise constant, and therefore this will give you $0$. The other option is to expand $\psi$ in terms of the energy eigenbasis by computing the coefficients $c_n = \langle \phi_n | \psi \rangle$, and getting $\langle H \rangle = \sum_{n=1}^\infty | c_n |^2 E_n$. As $E_n > 0$ for every $n$, this quantity will be strictly greater than $0$ and therefore will differ from the previous answer.</p>
<p>What is the discrepancy? It surely has to do with the discontinuity at $x = L/2$, but I can't figure out how to deal with it.</p>
| 9,508 |
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<p>I found the following picture in the Internet and I am curious how from a physicist point of view to explain it.</p>
<p><img src="http://i.stack.imgur.com/r33vQ.jpg" alt="enter image description here"></p>
<p>Basically the idea is the following. If you are a normal person - you suppose to see Einstein here, but if you are <a href="http://en.wikipedia.org/wiki/Myopia">myopic</a>, then you will see <a href="http://en.wikipedia.org/wiki/Marilyn_Monroe">Marilyn Monroe</a>. If you will move far away from a computer, you will still be able to see Monroe.</p>
<p>Can anyone explain it from a physical point of view?</p>
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<p>$$H_{tot}=\sum \dfrac{p_i^2}{2m}+\sum\dfrac{p_I^2}{2M_I}+\sum V_{nucl}(r_i)+\dfrac{1}{2}\sum_{i\ne j} \dfrac{e^2}{|r_i-r_j|}+\dfrac{1}{2}\sum_{I\ne J}\dfrac{z_Iz_Je^2}{|R_I-R_J|} $$</p>
<p>with:
$$V_{nuc}(r)=-\sum_I\dfrac{z_Ie^2}{|r-R_I|}$$</p>
<p>this equation is for the total non relativistic Hamiltonian of a system of electron and nucleus in mutual interaction via Coulomb forces. Here first term is for K.E. Of electron, second term is K.E for nucleus. My problem is on third term where negative sign is used. Is that potential energy of electron and nucleus? and in fourth and fifth term 1/2 is used. My question is why did 1/2 used in potential energy equation for both electron and nucleus?</p>
| 9,510 |
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<p>According to Biot-Savart Law, if there is a charged particle in motion, there will be a magnetic field. My question is whether the counterpart of this law also holds true, i.e. if there is a magnetic field, whether there will be a charged particle be in motion.
Let me explain my question a bit more clearly. First of all I don't know whether we can have a single charge in motion and another at rest in reality. If we can have then there may be two cases:</p>
<p>Case 1: At time t=0 a charged particle is placed at rest at point P where B=0 and at time t>0 an external B field is created in a region R covering the point P.</p>
<p>Case 2: At time t=0 a magnetic field B is created in a region R. At time t>0 a charged particle is placed at rest at point P in R. </p>
<p>Note of Caution: The point P is chosen in such a way that there won't be any external electric field at P.</p>
<p>Now come to the question: Will the charged particle move? If yes, under which case?</p>
<p>If 'NO', whether the charged particle will absorb any energy or not?</p>
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] |
<p>What is the difference between the <a href="http://www.google.com/#q=vernier+constant" rel="nofollow">Vernier constant</a> and <a href="http://www.google.com/#q=least+count+of+a+vernier+caliper" rel="nofollow">least count of a Vernier caliper</a>?</p>
| 9,512 |
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<p>For example, I am working on this problem, and I don't know where to begin. All the relationships that I can think of include a constant amplitude. [A (w/w^2) sin/cos theta]</p>
<p>Here's the problem:
A person rides on a mechanical bucking horse that oscillates up and down with simple harmonic motion. The period of the bucking is 0.74 s, and the amplitude is slowly increasing. What is the amplitude the rider must hang on to prevent separating from the mechanical horse?</p>
<p>Note: You don't have to give me the answer, just explain the appropriate relationships.</p>
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<p>I am trying to understand the differences between <a href="http://en.wikipedia.org/wiki/Wavefunction" rel="nofollow">wavefunction</a>, <a href="http://en.wikipedia.org/wiki/Probability" rel="nofollow">probability</a> and <a href="http://en.wikipedia.org/wiki/Probability_density" rel="nofollow">probability density</a>. There are different definitions on the internet. </p>
<p>For example:</p>
<p><img src="http://i.stack.imgur.com/Brj7x.jpg" alt="enter image description here">
<a href="http://inside.mines.edu/~fsarazin/phgn310/PDFs/QuantumLect4.pdf" rel="nofollow">http://inside.mines.edu/~fsarazin/phgn310/PDFs/QuantumLect4.pdf</a> </p>
<p>and</p>
<p><img src="http://i.stack.imgur.com/q6X4o.jpg" alt="enter image description here">
<a href="http://depts.washington.edu/chemcrs/bulkdisk/chem455A_aut05/hw_ans_nopass_wk%2004%20KEY.pdf" rel="nofollow">http://depts.washington.edu/chemcrs/bulkdisk/chem455A_aut05/hw_ans_nopass_wk%2004%20KEY.pdf</a></p>
<p>My questions are:</p>
<p>Which one is true?</p>
<p>What is the difference between three of them(wavefunction, probability and probability density)? </p>
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] |
<p>In quantum mechanics, any density matrix (or density operator) is Hermitian. Observables are also represented by Hermitian operators. So it follows that a density matrix can also be interpreted as an observable.</p>
<p>What is the physical meaning of this observable?</p>
| 9,515 |
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<p>I was working over some problems from my physics textbook, and I came across this one, it involves circular movement:</p>
<p>A car with the mass of 1t is moving over a hill with the velocity of 20 m/s. The radius of the hill (which can be viewed as a half-circle) is 100m.</p>
<p>When it comes to the top of the hill, with what force does it act upon the hill?</p>
<p>The correct answer is, apparently, 6kN. How do I solve this?</p>
<p>The situation can be illustrated like this I guess:
<img src="http://i.stack.imgur.com/zkukZ.png" alt="enter image description here"></p>
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<p>Is there light tunneling taking place in optically transparent mediums like glass? wherein light travels a larger path in these mediums without interacting with atoms and without any change in velocity, frequency, wavelength or momentum? but just getting affected by the field of atom and following the least resistant path through the medium?</p>
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<p>I thought of a thought experiment that had me questioning how <a href="http://en.wikipedia.org/wiki/T-symmetry" rel="nofollow">time reversal</a> works in <a href="http://en.wikipedia.org/wiki/T-symmetry#Time_reversal_in_quantum_mechanics" rel="nofollow">quantum mechanics</a> and the implications. The idea is this ... you are going forward in time when you decide to measure a particle. The particle then collapses to the observed state. Now if physics were to be the same in reverses time, then if we stop and reverse time then measure that very same particle again ... then I would imagine that since the wave function has collapsed we ought to measure the same thing. What this says to me is that given some time evolution in the + direction, if we measure a particle and it collapses the wave function, then if you reverse the arrow of time to go in the - direction we ought to get the same answer as before. The future/present effects the past. This means if we theoretically had a time machine and went back in time, we would have traveled into a different past.</p>
<p>Another implication of this thought experiment is that the future would be indistinguishable from the past and would hence forth be the same. I would imagine that this is consistent with the 2nd law of thermodynamics since physics dictates that entropy only increases ... going in the reverse direction of time to decrease entropy would violate the laws of physics. Has anyone else out there thought about this?</p>
<p>From my studies in quantum mechanics, I don't remember any postulates stating anything like this, but this all makes sense to me. Are there any theories out there that go along these lines?</p>
| 9,518 |
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<p>Aside from Noether's Theorem, how do we know energy is conserved?</p>
<p>Energy is the capacity of a system to do work. It's the number that tells me how much "force" a system can apply over a distance. For long I knew we could create forces. However, the only reason... The physical reason that conservation of energy is making sense to me is that when I apply a force to move matter from point $A$ to $B$, I must apply force $x$ on it over displacement $D$ in order to return it I must apply the same amount of force $x$ over $D$. Also the fact that all our natural forces are conservative (in most cases), if work is done by any of the four forces. You must apply the same work again because the natural force is acting against you. Example: Ball is attracted down to Earth's surface by gravity, $9.8$J of work is done ($mgh = 1 \times 9.8 \times 1$) $9.8$N of force has been applied. When the ball is at rest on the surface, I must apply the same amount of force against gravity to move it upwards back to point $A$ ($1$m above the surface). Does this make any sense?</p>
<p>Why is energy conserved besides Noether's Theorem? What other physical observations show it must be true in all our current observable systems? </p>
| 9,519 |
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<p>I am going through the idea of stabilizer formalism.</p>
<p>Defined what is a Pauli group $P_n$ and its properties, we describe a stabilizer set $S$ as:
$$S\subset P_n$$</p>
<p>The stabilizer set establishes valid codewords for a state if the equation $$s\left|\psi\right\rangle=\left|\psi\right\rangle,\;\;\;\forall s \in S \;\;\;\;\; (1)$$
is satisfied. That means $\left|\psi\right\rangle$ is a +1 eigenstate of $s$.</p>
<p>Each valid codeword belongs to $V$, that is a set of qubits stabilized by $S$. Therefore, if $(1)$ is satisfied, then $\left|\psi\right\rangle \in V$.</p>
<p>Let's consider the Steane code of 7 qubits.
The followings are the stabilizer codes for such encode: $$
K^1 = IIIXXXX
$$
$$
K^2 = XIXIXIX
$$
$$
K^3 = IXXIIXX
$$
$$
K^4 = IIIZZZZ
$$
$$
K^5 = ZIZIZIZ
$$
$$
K^6 = IZZIIZZ
$$</p>
<p>These reduce the $2^7$ Hilbert space into a two-dimensional subspace.
These stabilizers generate valid codewords for the Steane code: $$
\left|0\right\rangle_L \equiv \frac{1}{\sqrt{8}}(\left|0000000\right\rangle + \left|1010101\right\rangle + \left|0110011\right\rangle + \left|1100110\right\rangle + \left|0001111\right\rangle + \left|1011010\right\rangle + \left|0111100\right\rangle + \left|1101001\right\rangle)
$$
$$
\left|1\right\rangle_L \equiv \frac{1}{\sqrt{8}}(\left|1111111\right\rangle + \left|0101010\right\rangle + \left|1001100\right\rangle + \left|0011001\right\rangle + \left|1110000\right\rangle + \left|0100101\right\rangle + \left|1000011\right\rangle + \left|0010110\right\rangle)
$$</p>
<p>Here my doubt come; Each stabilizer is used as "filters of the input", so if an input, on which are applied one or more of those stabilizers, does not satisfy the equation $(1)$ ($\left|\psi\right\rangle$ -1 eigenvalue of $s$?), then we can say that an error occurred. Through <em>syndrome measurement</em> we can identify where the error occurred and correct it.</p>
<p>Another issue: verifying $(1)$ means, for example, $\;K^1 \left|1010101\right\rangle = \left|1011010\right\rangle$. Since both $\left|1010101\right\rangle$ and $\;\left|1011010\right\rangle$ represent $\left|0\right\rangle_L$, we say that $(1)$ is satisfied?</p>
<p>Finally: $\;K^4 \left|1010101\right\rangle = ?$</p>
<p>Thank you.</p>
<p>Added Last trouble:</p>
<p><img src="http://i.stack.imgur.com/Mj2gV.png" alt="Quantum circuit to prepare the [[7,1,3]] logical |0⟩ state"></p>
<p>The state of the system is represented by: </p>
<p>$$\left|\psi\right\rangle_F={1\over 2}(\left|\psi\right\rangle_I+U\left|\psi\right\rangle_I)\left|0\right\rangle + {1\over 2}(\left|\psi\right\rangle_I-U\left|\psi\right\rangle_I)\left|1\right\rangle$$</p>
<p>We apply $K^1,K^2,K^3$ to the input and we measure the ancilla qubits to verify the integrity of the input (if $\left|\psi\right\rangle_I$ is +1 eigenstate of $K^1,K^2,K^3$). If the equation $(1)$ is not satisfied, then the corrupted qubit is corrected with a $Z$ gate addressed by syndrome measurement of ancilla qubits. This is how does the system work?</p>
| 9,520 |
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<p>Can anyone demonstrate how to get the answers to these questions? I'm just interested in the method I need to use in order to obtain the correct answer no matter what the values are.</p>
<p>Three small spheres are placed at fixed points along the x-axis, whose positive direction points towards the right.</p>
<p>Sphere A is at x = 36.0 cm, with a charge of –8.00 μC.</p>
<p>Sphere B is at x = 41.0 cm, with a charge of 9.00 μC.</p>
<p>Sphere C is at x = 46.0 cm, with a charge of –3.00 μC.</p>
<p><strong>a)</strong> Calculate the magnitude of the electrostatic force on sphere B.</p>
<p><strong>b)</strong> Sphere B is now removed:
What would be the magnitude of the electric field at the point where sphere B was located?</p>
<p><strong>c)</strong> Sphere B is still missing. Give the x-coordinate of the point on the x-axis where the field due to spheres A and C is zero</p>
| 9,521 |
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<p>I have a airburst round with set amount of fragments with a certain shape inside my bullet. After a certain distance has travelled, a charge explodes in the back of the round disintegrating the shell and sending the fragments out in a forward cone.</p>
<p>My problem : </p>
<ol>
<li><p>Why do they shoot out in a cone? Or put in another way, why does a shotgun shell's fragments form a cone spread?</p></li>
<li><p>Would it be possible to choose a charge size/shape so that the discharge angle can be controlled? </p></li>
<li><p>Should every fragment be modelled separately after the explosion or can they be modelled as whole?</p></li>
</ol>
| 9,522 |
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<p>I would want to know what the differences between these two techniques in cristalography. I research on internet, but it is not clear to me what each one does. </p>
<ul>
<li>When it's better to use one and not the other? </li>
<li>What things that only one of them can give me and why? </li>
<li>In energy or security matter, there is some advantage using one or other?</li>
</ul>
<p>Thanks in advance.</p>
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<p>I'm looking for something short of a molecular dynamics package, where I can build up simple geometric shapes with flexible linkages/etc and simulate the consequences of electrostatic repulsion between surfaces. Something, say, that would let me simulate billiards in a magnetic field. </p>
<p>Does anyone have good recommendations for a 3D Newtonian dynamics software package (free or not) with these sort of capabilities? </p>
| 495 |
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] |
<p>I was wondering if it was possible to project a magnified image on a wall without the need of focusing, so just by dimensioning the lenses right.</p>
<p>I know I have to use the principal of Maxwellian View for the illumination of the slide. However, there are a lot of parameters left and I can design a system that requires a very minimal (say 1mm) displacement of the projection lens when projecting on 0.5m and, say, 2.5m. However, I was wondering if I oversee something here. I've seen that it is possible however I was also wondering if some special kind of lens is needed here.</p>
| 9,524 |
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<p>Can someone please explain the <a href="http://en.wikipedia.org/wiki/Sommerfeld_radiation_condition" rel="nofollow">Sommerfeld radiation condition</a> and what is the alternative non-reflecting boundary conditions for waveguides of general geometries?</p>
| 9,525 |
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<p>Can anyone explain to me why we have had no major breakthroughs in the theory of quantum computation in the past 15 years?</p>
<p>Shor's algorithm set the standard, since then we've had Grover's algorithm and an improvement on the Deutsch–Jozsa algorithm. What quantum algorithms of practical value have been created since Grover's?</p>
<p>What makes it so hard to create a quantum algorithm?</p>
<p>Thanks.</p>
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] |
<p>The <a href="http://www.iso.org/sites/JCGM/VIM/JCGM_200e.html" rel="nofollow">ISO VIM</a> defines them as:</p>
<blockquote>
<p><strong>measurement method:</strong>
generic description of a logical organization of operations used in a
measurement.</p>
<p><strong>measurement procedure:</strong> detailed description of a measurement according to one or more measurement principles and to a given measurement method, based on a measurement model and including any calculation to obtain a measurement result.</p>
</blockquote>
<p>But I'm still not sure what their difference is. Can anyone explain this to me, maybe with some examples?</p>
| 9,527 |
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<p>Currently accepted scientific theory says that all elements heavier than hydrogen, helium, and a little lithium have been created in supernova explosions.</p>
<p>My questions, specifically, is has anyone done the math to see if the observed amounts and the distribution of, the heavier elements in the universe agrees with the 13.7 billion year age of the universe?</p>
<p>With the birth, life, and death of stars at the currently observed and accepted rates... does the math work out? Can you please refer me to a resource where I can review the material or get further clarification. (I freely admit that my maths aren't up to the task, but I'm hoping for a deeper understanding of the phenomenon.)</p>
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] |
<blockquote>
<p>In an isothermal process the temperature of a system remains constant but may or may not be similar to temperature of surroundings</p>
</blockquote>
<p>How is this possible?</p>
| 9,529 |
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] |
<p>Non-collinear magnetism and paramagnetism, are they the same thing?</p>
| 9,530 |
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<p>Manganese has five unpaired electrons, but Iron has four, then why is Iron ferromagnetic and Manganese paramagnetic?
What's that property I'm missing?</p>
| 9,531 |
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<p>An elevator goes up 180 m by first accelerating at a constant rate of 1.0 m/s2, then staying at a constant
speed of 9 m/s and then decelerating at a constant rate of -‐‑1.0 m/s2. How much time does it take the
elevator to go from bottom to top?</p>
<p>so i know that d = 180m
a = 1.0 m/$s^2$ for the first part
v = 9 m/s
and for the last part a = -1 m/$s^2$</p>
<p>Im not sure which kinematics to use, there are two that I tried using</p>
<p>d=$v_i$t + 1/2a$t^2$</p>
<p>and</p>
<p>v$_f$ = $v_i$ + at</p>
<p>I tried to find the time for the part where the elevator is accelerating by doing</p>
<p>9 = 0 + 1t</p>
<p>so t = 9s</p>
<p>then I tried to find the distance for which the elevator was going up by doing</p>
<p>d = 0(9) + 1/2(1)(92) and got d = 40.5</p>
<p>so then the d for when its decelerating is 139.5 which i plugged into</p>
<p>d=$v_i$t + 1/2a$t^2$</p>
<p>and tried to solve for t but i ended up getting a weird answer with a square root.</p>
<p>where did i go wrong here?</p>
| 9,532 |
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] |
<p>scalars (spin-0) derivatives is expressed as:</p>
<p>$$\nabla_{i} \phi = \frac{\partial \phi}{ \partial x_{i}}$$</p>
<p>vector (spin-1) derivatives are expressed as:</p>
<p>$$\nabla_{i} V^{k} = \frac{\partial V^{k}}{ \partial x_{i}} + \Gamma^k_{m i} V^m$$</p>
<p>what is the expression for covariant derivatives of spinor (spin-1/2) quantities?</p>
| 9,533 |
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<p>I have heard the term quantum leap to describe the (instantaneous?) transition from a higher energy orbital to a lower energy orbital. Yet, I understand that this transition time has now been quantified in certain experiments.</p>
<p>So, it is not a leap? Is the electron wavefunction defined during this transition?</p>
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<p>If I have a positive charge of +q and a negative charge of -q that are set a distance of r apart from each other, their midpoint will have an electric potential of zero. However, if I put a test charge into this midpoint and release it, it will gain kinetic energy, and I am under the impression that it comes from some sort of potential energy. But according to my book, that midpoint that it was placed at has no potential? Where does the kinetic energy come from? </p>
| 9,535 |
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<p>Assume there is an infinite grounded conducting plane in the $xy$-plane. Assume also that a point charge is being brought slowly from $z=\infty$ to $z=z_0$, slowly enough that the problem is treated electrostatically. This charge will induce charge distribution on the plane. Now, I want to know why the work done by the point charge on the charges on the plane is zero. </p>
<p>I know that at equilibrium the field lines at the conductor are perpendicular to the plane. But instantaneously, as I move the charge, this is no longer the case, and in a very short time, equilibrium is reached, and the lines are perpendicular. But this was done at a cost of work.</p>
| 9,536 |
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<p><img src="http://i.stack.imgur.com/IDIYj.jpg" alt="Picture"></p>
<p>a mass hangs from three ropes (as shown in the Picture)</p>
<p>can someone pleas help me with this?</p>
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] |
<p>According to our current scientific knowledge, how certain is it that <a href="http://en.wikipedia.org/wiki/Heat_death_of_the_universe" rel="nofollow">heat death</a> shall be the ultimate fate of our universe, and why? Are there any serious hypotheses competing with heat death, and if such hypotheses exist, what are they based on?</p>
| 9,538 |
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<p>A particle's interaction (with anything it can interact with) can be thought of as it making a measurement of the physical quantity associated with the interaction, (e.g. electric field in case of the interaction between charged particles) and acting accordingly. To make a measurement one first needs a frame to make measurements "in" (a Lorentz frame in light of relativity). Assigning such frame to a photon appears to me, to be problematic in the sense that constant velocity of a photon in any inertial frame implies a photon in its own frame having a velocity $c$ (the speed of light). Now if we assume that no such frame exists for a photon, photons cannot interact with an other one. </p>
<p>Can we explain in this way that photons do not interact with each other, or more generally particles moving at velocity $c$ do not interact with similar particles?</p>
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] |
<p>Phase space flow shares characteristics with fluid flow such as incompressibility by Liouville's theorem. Extending the similarities one might be curious, does phase space flow have a characteristic number like the Reynold's number? Moreover, can phase space flow exhibit the characteristics of turbulence? </p>
<p>If so, can someone suggest papers or text concerning the aforementioned questions. </p>
| 9,540 |
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] |
<p>The common definition of "time" is a type of measurement, like size. But the sentence "size gets bigger" doesn't make any sense. Is "time slows down" an odd phrasing of "events occur more slowly" or is there a deeper meaning to the phrase?</p>
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<p>Let's have interaction between gravitational and scalar real fields. For an action of gravitational field in vacuum I add term $S_{m} = \int d^{4}x\sqrt{-g}L_{m}$, where
$$
L_{m} = \frac{1}{2}g^{\mu \nu}\partial_{\mu}\varphi \partial_{\nu} \varphi - V(\varphi )
$$
So the variation $\delta S_{m}$ must give $\frac{1}{2}\int d^{4}x \sqrt{-g}\delta (g^{\mu \nu})T_{\mu \nu}$, where $T_{\mu \nu}$ refers to the stress-energy tensor of scalar real field. I tried to do it, but this leads me to the answer
$$
\delta S_{m} = \int d^{4}x\delta (\sqrt{-g})L_{m} +
$$
$$
+\int d^{4}x \sqrt{-g}\left(\frac{1}{2}\delta (g^{\mu \nu})\partial_{\mu} \varphi \partial_{\nu} \varphi - g^{\mu \nu} \partial_{\mu}\varphi \partial_{\nu} \delta \varphi - \frac{\partial V( \varphi )}{\partial \varphi }\delta \varphi\right)=
$$
$$
=\frac{1}{2}\int d^{4}x\sqrt{-g}\delta (g^{\mu \nu})T_{\mu \nu} + \int d^{4}x\sqrt{-g}\left( g^{\mu \nu} \partial_{\mu}\varphi \partial_{\nu} \delta \varphi - \frac{\partial V( \varphi )}{\partial \varphi }\delta \varphi\right).
$$
What to do with the second integral? Does it equal to zero according to Euler-Lagrange equation, or I can't say so?</p>
| 9,542 |
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] |
<p>I read in a book just now that says:</p>
<blockquote>
<p>[...] but it was not explained until the invention of quantum field theory by Richard Feynmann [<a href="http://en.wikipedia.org/wiki/Sic"><em>sic</em></a>] and others in the 1940's.</p>
</blockquote>
<p>I have been under the impression that it was Paul Dirac and Hermann Weyl in the early to mid thirties who developed the foundations of QFT.</p>
| 9,543 |
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