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What is quantum discord? What is quantum discord? I stumbled upon this term on Quantum Computing: The power of discord, but have never heard of it before. Can you give a bit more mathematical explanation of the term here?
| It is basically a measure of the quantumness of some correlations, which is not vanishing for some separable state. It was introduced by Ollivier and Zurek (PRL/arXiv). It is the difference between two different generalizations of the classical (Shannon) conditional entropy to the quantum world, and is 0 for a pure bipartite separable state. It has been proven to be the amount of entanglement needed in the task of state-merging (PRA/arXiv and PRA/arXiv).
Definition
(PRL/arXiv)
Classically the conditional entropy $H(A|B)$ is a measure of the uncertainty one has on the variable $A$ once we know the variable $B$. Of course, the definition of "knowing" $B$ becomes problematic when $B$ is quantum.
*
*Classically, one can define $H(A|B)$ as the average $H(A|B)=\sum_b {\mathcal P}(B=b)H(A|B=b)$, each $H(A|B=b)$ being the entropy of $A$ given that the random variable has the value $b$. If one generalizes this to the quantum world, the $B=b$ part implies a quantum measurement (a POVM) which should be specified. A natural choice is the "best" measurement, the one which minimizes the entropy. The Shannon $H$ entropy is replaced by the Von Neumann entropy, and we define
$S(A|B_c)=\min_{\text{POVM}} \sum_{b}\mathcal{P}(\text{POVM applied to B gives } b) S(A|\text{POVM applied to B gives }b)$.
*The previous definition leads classically to a redefinition of the conditional entropy
as an entropy difference : $H(A|B)=H(A,B)-H(B)$, which is always positive. Its quantum version, $S(A|B)=S(AB)-S(B)$ can be negative (in contrast with $S(A|B_c)$). Its negativity is a sufficient condition for entanglement.
The discord is defined as $S(A|B_v)-S(A|B)$ and is always positive. You can maybe see it
as the amount of correlation between $A$ and $B$ which is destroyed by a classical measurement of $B$.
Link with state merging
(PRA/arXiv and PRA/arXiv)
The state merging primitive is the following. Suppose Alice, Bob and Charly share a 3-party pure entangled state. Alice want to send her part to Bob without destroying the quantum correlations between $AB$ and $C$. Basically, she has to teleport $A$ to Bob, and the minimal amount of entanglement Alice and Bob need to perform this task is given by the quantum discord.
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Can quantum annealing be used for factorization? It is known that there is a famous quantum factorization algorithm by Peter Shor. The algorithm is thought to be suitable only for quantum gate computer.
But can a an adiabatic quantum computer especially that which is capable of quantum annealing be used for factorization?
I am asking this because it seems that Geordie Rose claims in his blog that they have a quantum factorization algorithm that is somehow "better than Shor". But the details are unavailable as of now.
| Yes, though I don't think that we'll see D-Wave factoring even 20-bit numbers anytime soon. One of their tutorials shows how to model a NAND gate using 4 qubits. With a handful of those, I can make a carry-save multiplier cell, though surely it can be built more optimally. If I want to factor an N-bit number, I could use an N/2 by N/2 array of the carry save adder cells, and constrain the N-bit output to equal the number I want to factor, and have no weights on the inputs. Run Quantum Annealing, and in theory with probability approaching 100% as noise goes to zero and run time get's longer, and the inputs will settle to the input factors, in one of the two acceptable states, for example 3x5 = 15 vs 5x3 = 15.
The title "Better than Shor" may simply mean that with their new 512 qubit QAO, they believe they can factor 35 = 5x7, or maybe 51 = 3x17. I really don't see factoring a 512 bit number with a 512 qubit quantum annealer. Since building the multiplier takes O(N^2) qubits, we'll probably need over a million to factor 2048 bit RSA. A modified Booth Encoded multiplier saves you a factor of over 2. If D-Wave continues doubling qubits every year or so, and if they continue showing true quantum annealing performance, we may need to switch to a post-quantum-computer encryption algorithm.
Note that this technique also works for finding SHA-1 collisions. It's super cool stuff. I just found a paper describing the algorithm from 2002: http://arxiv.org/abs/quant-ph/0209084
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Mechanical work to required battery power I have a very practical question where I've calculated the mechanical work needed by a simple mechanical system by solving the line integral $W = \int_C \ F \ dx$. However, since I have a black spot in my brain for electrical calculations I figured I could just (considering a 1.5V 2700mAh Alkaline battery) calculate the current required by $I = P / V$ and using that calculate how many batteries I need or how long my mechanical system could work. Is this correct or am I missing something crucial here?
| From the specifications of your battery, that is 1.5V and 2700mAh, you can compute that there is $14580$ Joules of energy stored in your batteries.
The formula $P=U\cdot I$ relates power to voltage and current. You battery specs give voltage and capacity (that is total charge stored). The former is in Volt, the latter in milli-Ampère-hour. The product is therefore not a power but an energy. And that corresponds to the $14580$ Joules I mentioned. Of course, you should also factor in possible losses, so in practice, you'll never draw that much work from the batteries.
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Analyzing the motion of a ball rolling without slipping inside a hemispherical bowl Consider a solid ball of radius $r$ and mass $m$ rolling without slipping in a hemispherical bowl of radius $R$ (simple back and forth motion). Now, I assume the oscillations are small and so the small angle approximation holds. I wish to find the period of oscillation and I analyze the motion in two ways, first using conservation of energy and secondly using dynamics. However, I receive two inconsistent answers. One or both of the solutions must be wrong, but I cannot figure out which one and more importantly, I cannot figure out why.
Method 1: We write the energy conservation equation for the ball
$mgh + \frac{1}{2}mv^2 + \frac{1}{2}I\omega^2 = Constant$
from the center of mass, we take the height as $h = R-(R-r)cos\theta$ where $\theta$ is the angle from the vertical. Applying the no slip condition $v = r\omega$ and taking the moment of inertia for a solid sphere $I = \frac{2}{5}mr^2$ we can write the energy equation as
$mg(R-(R-r)cos\theta) + \frac{7}{10}mr^2\omega^2 = Constant$
Differentiating with respect to time:
$mg(R-r)sin\theta\cdot\omega + \frac{7}{5}mr^2\omega\cdot\alpha = 0$
taking the small angle approximation $sin\theta = \theta$ we get
$g(R-r)\theta + \frac{7}{5}r^2\alpha=0$
$-\frac{5g(R-r)}{7r^2}\theta = \alpha$
from which we can get $T = 2\pi\sqrt{\frac{7r^2}{5g(R-r)}}$
Method 2: The only torque acting on the ball at any point in its motion is the friction force $f$. So we can write
$\tau = I\alpha = fr$
again using the rolling condition $a = r\alpha$ and the moment of inertia for a solid sphere,
$\frac{2}{5}ma = f$
The net force acting on the system is the tangential component of gravity and the force of friction, so
$F = ma = mgsin\theta - f$
$\frac{7}{5}a = gsin\theta$
taking the small angle approximation and converting $a$ to $\alpha$ we get
$\alpha = \frac{5g}{7r}\theta$
and a corresponding period of $T = 2\pi\sqrt{\frac{7r}{5g}}$
Now the solutions are very different and I would appreciate it if someone would point out where I went wrong.
| Your first derivation, using energy, uses two different meanings for the same symbol $\omega$. In one place, you interpret it as
$$\omega = \dot{\theta}$$
the time derivative of the angle of the line from the center of the ball to the center of the bowl with the vertical.
In another place, you interpret $\omega$ as the time derivative of the unnamed angle through which the ball itself has rotated.
These two angles are related to each other by the $r/(R-r)$ factor by which you are off.
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what cools bottle of water faster: ice or snow Imagine you have a pile of snow and a pile of ice shards. You put a soda bottle which has a room temperature into both piles. Which bottle is going to cool down faster?
| The first thing to realize is that "ice" is a pile of small ice shards and snow is a pile of itty-bitty ice shards.
Assuming the snow and ice are at the same temperature, the answer to your question with come down to which one has more contact area and (to a much lesser extent) how that contact area is distributed. Also note that the contact area and its distribution could change over time, as the can melts the ice/snow.
You'd have to run the experiment. My guess is that ice would be the winner. This is because ice would maintain a much larger contact area over time. The can in the snow would melt the snow beneath it and drop away from the "snow roof" over it; whereas I would expect the "ice roof" to collapse with the can, continuing to cool it. The can contents in contact with the roof would be expected to be the warmest, thereby giving maximum cooling effect from any contact on the top side of the can.
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Ascent rate and size of balloon I am part of a school project, Project Stratos to send a balloon to the edge of space (the closer side :P) and was wondering how you would work out the accent rate of a large balloon (roughly 1m^3 of helium with 100g of mass) and the size of it as it increases its Altitude. I am creating a live map (that will be based on predictions rather than its actual location) and want to know the speed it will float up into the atmosphere. Currently we are assuming the ascent rate will be about 5m/s but I doubt that is very accurate and would this speed increase as it gets higher?
Edit: I would also quite like to know the burst height of the balloon.
| Your answer using drag assumes the balloon is rising very fast. At the low speeds, typical of a balloon, the dominant resistive force is the viscose downward displacement of the surrounding gas as the balloon rises upwards. This depends linearly on speed, not quadratically. Think lava lamp, not fighter jet.
The laboratory test for this, that can be done in any elementary school, would be to measure the speed of rise with different weights attached to the same helium balloon. One will find the rate of rise drops linearly as the effective density of the balloon (balloon + weight) increases towards atmospheric density. More importantly, the rate of fall, will increase if the density of the balloon is greater than the atmospheric density. This points to the other flaw in your reasoning, in your equation the direction of motion is always positive, or at least undefined, if the balloon is more dense than the air.
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Direct exposure to the vacuum of space I was watching a few sci-fi movies and was wondering the real science explaining what would happen if you were to be subject to the conditions of outerspace.
I read the wikipedia article on space exposure, but was still confused. If a person was about the same distance from the sun as earth is, would they still freeze to death? (as shown in the movie Sunshine)
I'm reading from all sorts of sites with conflicting information about what would actually happen when a person is exposed to the vacuum of space...
| You'd freeze to death faster in the Atlantic ocean.
Space has essentially no thermal conductivity. All the heat you lose will be radiated away. According to the Stefan-Boltzman law, $W = \sigma T^4$, you would lose at most 500 watts per square meter of body surface area. By contrast, the convective heat transfer coefficient in water is about 12,500 watts/square meter / degree Kelvin temperature difference. So, I think freezing would be the least of your concerns.
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What other shielding material than lead is effective against gamma rays? As the question in the title states I am wondering what material can be effectively used to shield gamma rays apart from lead? I believe concrete is often used, but it is nowhere near as effective as lead (6 cm to match 1 cm of lead as I understand it). I also hear significant bodies of water helps, as does tightly packed dirt, but surely there must be other materials that shield nearly as effectively as lead?
| There are three processes by which gamma rays interact with matter: the photoelectric effect, Compton scattering, and pair production.
The photoelectric effect is an interaction between the gamma ray and an electron. It's forbidden by conservation of energy and momentum unless there is some other body present as well, such as an atomic nucleus. For this reason, the probability of the photoelectric effect is proportional not just to the density of electrons but also approximately to $Z^n$, where $Z$ is the atomic number and $n$ is about 4 to 5.
Compton scattering can occur without the presence of anything besides an electron, so it only depends on electron density.
Pair production goes like $Z^2$ at typical gamma-ray energies.
For these reasons, the best shielding against gamma rays is achieved with a substance that has a high density of electrons (which correlates with a high mass density) and also a high $Z$. Lead has these properties. It's also cheap. There are elements with a higher $Z$, such as bismuth, polonium, and uranium, but they aren't cheap, and their atomic numbers are only slightly higher.
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Why is the gravitational force always attractive? Why is the gravitational force always attractive? Is there another way to explain this without the curvature of space time?
PS: If the simple answer to this question is that mass makes space-time curve in a concave fashion, I can rephrase the question as why does mass make space-time always curve concavely?
| Artwork:
dipole +- <--- some distance ---> +- dipole
Two dipoles are always attractive (or a dipole and another charge).
If they are like this +- ... -+ or -+ ...+- the dipoles will rotate and the configuration become attractive +- ... +- or -+ ... -+ .
They obey a 1/r³ relation.
If you can consider that inside the baryons (neutron, proton) can exist a configuration of dipoles you have an answer. (read the book of Douglas Pinnow, 'Our Resonant Universe'. It is a monography of a model of particles where this happens).
How do we go from 1/r³ to 1/r²? Integrate along the path.
Why? Explore the concept of polarizable vacuum.
The consensus is that gravitation is not electromagnetism, but in that way it is always attractive. And I like it.
But the order of magnitude of gravitation is $10^{-35}$ (more or less, by memory) of EM, so how can it be EM?
Can you figure out two EM radiators in each dipole in opposition of phase (one the +, other the -) extremely near one of the other? Yes, the radiated EM field has to be extremely faint.
(The connection with EM is much more compelling, IMO, that a connection with thermodynamics or other...)
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Why so many arguments for the transformation equations of generalized coordinates? For a system of $N$ particles with $k$ holonomic constraints, their Cartesian coordinates are expressed in terms of generalized coordinates as $$\mathbf{r}_1 = \mathbf{r}_1(q_1, q_2,..., q_{3N-k}, t)$$ $$...$$ $$\mathbf{r}_N = \mathbf{r}_N(q_1, q_2,..., q_{3N-k}, t)$$
Each particle in space can be uniquely identified by 3 independent variables, so why aren't the above of the form $$\mathbf{r}_i = \mathbf{r}(q_{i1}, q_{i2}, q_{i3})?$$
Note there is only one transformation $\mathbf{r}$ for all $\mathbf{r}_i$, a function of only three generalised coordinates and independent of $t$.
| The $k$ holonomic constraints are used to eliminate $k$ $q$s, so reducing their number from $3N$ to $3N-k$. This then introduces the dependence of some of the transformation equations on t and other $q$s.
You have k holonomic constraints of the form $$\mathbf{f}_1(q_1, q_2,..., q_{3N},t) = 0$$ $$...$$ $$\mathbf{f}_k(q_1, q_2,...,q_{3N}, t) = 0$$ $3N$ q coordinates for the $N$ particles $$(q_1,q_2,q_3), (q_4,q_5,q_6),..., (q_{3N-2},q_{3N-1},q_{3N})$$ $3N$ transformation equations relating cartesian to generalised coordinates $$\mathbf{r}_1 = \mathbf{r}_1(q_1,q_2,q_3)$$ $$...$$ $$\mathbf{r}_N = \mathbf{r}_N(q_{3N-2},q_{3N-1},q_{3N})$$ Using the first constraint to eliminate $q_1$ gives $\mathbf{r}_1= \mathbf{g}_1(q_2,q_3,..,q_{3N},t)$ which is of the same form as one of the transformation equations you quoted. Which $q$ can be eliminated depends upon the constraints of the problem and so in general, the transformation equations are of the form $$\mathbf{r}_i= \mathbf{r}_i(q_1,q_2,..,q_{3N-k},t)$$
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How is thermodynamic entropy defined? What is its relationship to information entropy? I read that thermodynamic entropy is a measure of the number of microenergy states. What is the derivation for $S=k\log N$, where $k$ is Boltzmann constant, $N$ number of microenergy states.
How is the logarithmic measure justified?
Does thermodynamic entropy have anything to do with information entropy (defined by Shannon) used in information theory?
| Hey you have the Thermodynamic definition which is :
$$\Delta S = \int \frac{dQ}{T}$$
and the statistical definition:
$S=k\log N$
N- the number of the possibly states of the system (including degeneration).
For mostly everything they are equivalent!!
note that you can not measure enytopy, but you can measure the $\Delta$ (the change) in the entropy of the system.
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In a gas of particles, how is the displacement vector related to the number density? Suppose I have a gas of particles that is initially uniformly distributed so that the number density is $n_0$ (number of particles per unit volume), and then I displace the particles by the vector field $\vec{d}(\vec{x})$ (i.e. the particle initially at position $\vec{x}$ is displaced by the vector $\vec{d}$). How is the resulting number density $n(\vec{x})$ related to the displacement vector $\vec{d}(\vec{x})$?
I'm sure this must be done somewhere in a standard textbook but I can't find where.
| A simple 1D calculation gives, at first order, something like
$$
\frac{1}{n} = \frac{1}{n_0}\left(1+\vec{\nabla}\cdot\vec{d}\right)
$$
but only if $\vec{d}$ is small enough. Otherwise, for calculating
$n(\vec{x})$, you need to evaluate the divergence at a point $\vec{x'}$
such that $\vec{x'}+\vec{d}(\vec{x'}) = \vec{x}$.
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Linear polarizer and the angle of incidence It is known that when a beam of lineary polarized light falls perpendicularly on a linear polarizer, the intensity of polarization changes according to Malus' law and the direction of polarization changes as cosine of angle between polarization vector and polarizer vector.
My question is: is anyone familiar with mathematical treatment of how the direction and intensity of polarization changes when the angle of incidence changes?
| It depends on the polarizing material.
I assume below that it is about layer of absorptive material.
When wave travels through an anisotropic material, its phase shift and absorption is polarization-dependent. So you should be able to treat it in similar way as the Birefringence (giving a complex refraction index to take into account the absorption).
If you want to consider a simple case of very thin polarizer, with the 100% absorption of one polarization, and 0% of the another - it simplifies. Just you need to say if polarization in the direction $z$ (i.e. the perpendicular to the polarizer) is absorbed or not. Of course for other efficiencies it is still relatively simple, as long as you study the case of the very thin polarizer.
If you are interested in analyzing different polarizers (e.g. polarizing cubes, basing on the birefringence and total internal reflection - then you can use the Fresnel equations).
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A No-Nonsense Introduction to Quantum Field Theory I found Sean Carroll's "A No Nonsense Introduction to General Relativity" (about page here. pdf here), a 24-page overview of the topic, very helpful for beginning study. It all got me over the hump of learning the meaning of various terms associated with GR, most of which I had heard before without understanding. It also outlined the most important examples.
Is there a similar document for quantum field theory, which presents the main equations, briefly describes the main ideas, and summarizes the most important applications and results so that the reader can feel the lay of the land before studying in depth?
| There is a great introduction called "This is How Quantum Field Theory Works" which, I think, is exactly what you are looking for.
All essential concepts are introduced and the basic idea how one gets from the fundamental equations to cross sections, i.e. quantities that can be measured in experiments is sketched.
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Why aren't there compression waves in electromagnetic fields? I just started learning about optics, and in the book I'm reading they explain how the electrical field caused by a single charged particle could be described by a series of field lines radiating out of the particle; they compare these imaginary lines to ropes, to provide an intuition of the concept.
Then they say that and that if we wiggle the particle up and down, that would produce transversal waves in the horizontal field lines, but no waves in the vertical lines. I know that the physical analogy is not to be taken literally, but I don't understand why wouldn't that cause compression waves in the vertical lines.
I mean, even though the direction of the field in the areas directly above and below the particle doesn't change, the intensity does. And I assume it wouldn't instantly. So what am I missing?
| Well, is not 100% percent accurate to say that there aren't longitudinal EM waves. In a waveguide there are allowed propagation modes that have non-zero electric and magnetic components in the direction of propagation:
http://en.wikipedia.org/wiki/Transverse_mode
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Photon statistics of an incandescent light source We usually calibrate the cameras on our microscopes by capturing 20 images of a blurry (not sharp) fluorescent particle. For each pixel in this stack of 20 images we calculate the intensity variance. By plotting the variance against the intensity we obtain the scale factor that allows us to convert the ADU (arbitrary data unit) into detected photoelectrons.
However, this approach relies on a light source with Poisson statistics.
This is applicable for fluorescence light.
But can we also use light of a halogen lamp instead of fluorescent light?
I heard that photons leaving a heated metal surface have a particular statistic. I think the argument was that they are coming out in bunches and it has something to do with the workfunction. It would be nice if someone could give a good reference on this.
The picture above shows the calibration curves for an EMCCD camera.
The left graph corresponds to the readout mode you would use for bright images and has a readnoise of 15 electrons per pixel.
The graph on the right shows the result for the low noise mode. In this mode the charge that has been accumulated under the pixels is transferred through an electron multiplication register. Impact ionization increases the number of electrons. The gain is usually up to 300x.
The inset on the lower right is how a calibration image looks like. Note that this one is fake.
| I noticed this old answer and I don't think it's right. Surely a laser gives off photons with poisson statistics, but not an incandescent source. The reason is fairly intuitive, to the extent that these types of verbal arguments are ever correct: there are random intensity fluctuations associated with incandescent sources; so if there is a click in the detector, there is an enhanced probability that it occurs during an intensity excursion. Therefore, a second click very close by is more likely than average. This is in contradiction to the definition of Poisson distribution, which is simply that the probability of clicks is independent of the proximity to any other click.
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Calculating time for a fully charged UPS I have a UPS of 1000 Volts connected with 2 batteries each of 150 Amp. How much time it will take to consume the whole UPS (after fully charged) when a device of 1Amp is getting electricity form that UPS.
Please also explain me the calculation.
| Don't think you can calculate this. I depends on the type, age and history of the batteries. All his factors make a big difference. If you rely need to know the TCT "total charge time" put them threw the cycle once and measure it. And remember this will change with time.
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Why should one expect closed timelike curves to be impossible in quantum gravity? From the Wikipedia article, it seems that physicists tend to view closed timelike curves as an undesirable attribute of a solution to the Einstein Field Equations. Hawking formulated the Chronology protection conjecture, which I understand essentially to mean that we expect a theory of quantum gravity to rule out closed timelike curves.
I am well-aware that the existence of closed timelike curves implies that time travel is technically possible, but this argument for why they should not exist isn't convincing to me. For one, if the minimal length of any closed timelike curve is rather large, time travel would be at least infeasable. Furthermore, this is essentially a philosophical argument, which is based, at least in part, on our desire to retain causality in studying the large scale structure of the universe.
So far, the best argument I've heard against CTCs is that the 2nd law of thermodynamics wouldn't seem to have a meaningful interpretation in such a universe, but this isn't totally convincing. A good answer to this question would be some form of mathematical heuristic showing that in certain naive ways of combining quantum mechanics and gravity, CTCs are at least implausible in some way. Essentially, I'm trying to find any kind of an argument in favor of Hawking's conjecture which is not mostly philosophical. I realize that such an argument may not exist (especially since no real theory of quantum gravity exists), so other consequences of the (non)existence of CTCs would be helpful.
| It's certainly possible that just as the holographic principle can save unitarity for black holes, that some generalization of the holographic principle coupled with cosmic censorship for closed timelike curves can save unitarity.
At any rate, the interior of a time machine is only real to the extent that memories and records of the interior can get out.
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Gelfand-Yaglom theorem for functional determinants What is the 'Gelfand-Yaglom' Theorem? I have heard that it is used to calculate Functional determinants by solving an initial value problem of the form
$Hy(x)-zy(x)=0$ with $y(0)=0$ and $y'(0)=1$. Here $H$ is the Hamiltonian and $z$ is a real parameter.
Is it that simple? If $H$ is a Hamiltonian, could I use the WKB approximation to solve the initial value problem and to be valid for $z$ big?
| I was at a talk a while back by Gerald Dunne where he talked about the Gelfand-Yaglom theorem. He used it for calculating some Euler-Heisenberg type effective actions. A paper of his with Hyunsoo Min on the subject is A comment on the Gelfand–Yaglom theorem, zeta functions and heat kernels for PT-symmetric Hamiltonians and he's got some nice lecture notes: Functional Determinants in Quantum Field Theory (also see a wider spanning set of lectures of the same name).
Basically, it's a way of calculating the determinant of a 1-dimensional operator $\det(H)=\prod_i \lambda_i$ with out calculating, let alone multiplying, any of its eigenvalues $H \psi_i = \lambda_i \psi_i$.
To state the original theorem: assume that you have a Schrodinger operator (or Hamiltonian)
$ H = -\frac{d^2}{d x^2} + V(x) $
on the interval $x\in[0,L]$ with Dirichlet boundary conditions:
$$ H \psi_i(x) = \lambda_i \psi_i(x) \,,
\quad \psi(0)=\psi(L)=0 \ .
$$
Then we can compute its determinant by solving the related initial value problem
$$ H \phi(x) = 0\,, \quad \phi(0)=0\,,\quad \phi'(0) = 1 \ ,$$
so that
$$ \det H \approx \phi(L) \,,$$
where the final result is only $\approx$ as we can only really calculate the ratio of two determinants.
This basic result can be generalised to more general boundary conditions, coupled systems of ODEs and higher order linear ODEs.
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What happens to light in a perfect reflective sphere? Let's say you have the ability to shine some light into a perfectly round sphere and the sphere's interior surface was perfectly smooth and reflective and there was no way for the light to escape.
If you could observe the inside of the sphere, what would you observe? A glow? And would temperature affect the outcome?
Seems silly, it's just something I've always thought about but never spent enough time (until now) to actually find an answer.
| As soon as the light shining in was turned off, the light in the sphere would disappear, not because observing depletes energy, it doesn't (but energy loss by the system is required for observation to occur). No one knows but this hypothesis that light beams persist when the light emitter is extinguished is just that. There is no supportive evidence for this theory, that I am aware of.
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Simplest interferometer I want to build simplest interferometer which should be able to measure movements down to fraction of wavelength.
What is the simplest scheme for that, and what are the requirements for a laser?
I have a bunch of laser diode-based ones, and I guess they might be not coherent enough...
Are green DPSS ones any better?
| You should also consider the laser's frequency stability. My understanding is that the frequency of light can go up and down by some nanometers as temperature (and perhaps some of the electrical inputs) of a laser diode varies. Searching seems to turn up quite a few articles on stabilizing their output, exactly for this purpose.
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Projecting image without manual focussing 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.
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.
| You can do it by correctly choosing the focal length of the lens(es) you're using. Is that what you mean? The drawback is that you can only use it in one specific setup. (I'm not sure if I understand correctly as this seems a bit trivial).
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How fast does an ice cube melt in a microwave? I have noticed that when I microwave an ice cube it appears to melt more slowly than I would expect. For example, an equal volume of water starting at 0 deg C would probably be at boiling point before an ice cube that was at -15 deg C had melted. I realize there is enthalpy of fusion to take into account in the melting process but I believe there is more to it than that.
As I understand it a microwave oven works by exciting the water molecules in whatever is being cooked and if memory serves the frequency used is one that causes rotation of the molecule. Since the ice cube is solid I'm assuming the molecules aren't free to rotate and therefore the microwaves have a much reduced effect. In fact I'm wondering if a perfect single crystal of water would respond at all to being microwaved. Does this sound right?
I've been trying to rack my brain for a way of testing this theory but I can't think of a way of getting an perfectly dry ice cube into a microwave to see if anything happens. Even a tiny amount of surface water, caused from interaction with a warm atmosphere, would encourage melting.
| The unusual thing is the really high absorption of microwaves by
bulk water, whereas the ice behaves more normal like most solids and liquids.
In liquid water we have an effect of relaxation of orientational polarisation.
The polarisation is achieved not by rotation (not possible in liquid water)
but by shift of hydrogen atoms along the hydrogen bonds.
This is a kind of Kohlrausch conduction mechanism.
This process is extremely fast, so polarisation of water is
one of the fastest processes in liquids.
There is debate, whether tunneling plays a role to enhance the shift of the protons.
The same mechanism is responsible for the extraordinary (about tenfold)
mobilities of H+ Ions in water.
Here is a plot of Water spectrum in microwave domain. Note the incredible
absorption maximum with k about 3 !
http://books.google.com/books?id=bj1EnQPB0CMC&pg=PA184#v=onepage&q&f=false
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Can an object between two magnets reduce their attraction/repulsion? If you have two magnets (not in contact) and then put a non-magnetic object in between the two magnets, does that decrease the attraction/repulsion between the two magnets?
It seems that it wouldn't, because, if it would, then you could allow two magnets to attract each other, then put an object in between before they came in contact, and then be able to pull them apart again without having to put in as much energy as you would have to if the object had not been put in; you would get "free energy" (in the sense of a "free lunch").
Can anyone confirm this?
| Of course it's possible and it's completely the same like with electric materials. Google for terms (electric and magnetic) polarization, permitivity, permeability, susceptibility.
The problem with your perpetum mobile is that you don't account for bringing in the third object at all.
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How beam focusing looks like in electron microscope? I mean I know there are electrostatic/electromagnetic lenses which does focus the beam, but I am not sure how it is possible to foсus beam down to a few 10nm while emitter might be 1mm thick while having large focusing distance (especially when looking at chromatic aberrations..) - optics approach says you would not be able even get 1mm spot at such NA...
Why doesn't electrons widen the beam by repelling each other en-route?
Is there any good books on the subject explaining all these tricky stuff of focusing electron beam?
| "Why doesn't electrons widen the beam by repelling each other en-route?"
This does happen and can cause unwanted distortions. The reason it's not an overwhelming problem is that the electrons, once they get going, are not all that close together. For example, if you have a 1nA beam current, and the electrons are traveling near the speed of light, there is on average 5cm vertical distance between one electron and the next. Also, once they're traveling very fast, it won't be very long until they to reach the sample, so even if they're slightly pushing each other, they can't get very far off track.
Everything gets worse when the beam current goes up, and also when the electron energy (hence speed) goes down. More specifically regarding electron energy: (1) When you lower the beam voltage, repulsion effects will get worse; (2) The beam path very close to the electron gun (before the electrons have accelerated much) is the part that needs to be designed most carefully to minimize repulsion.
I'm not an expert, this is my imperfect recollection from a few years ago. :-)
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Finding the position of a planet between two other planets of known mass and distance Here is the question:
A planet with mass $m$ and a second with mass $M$ are separated by a
distance $d$. A third planet with mass $m_3$ happens to be midway
between $M$ and $m$. Where could the third planet be positioned
(distance from the larger planet $M$ in meters) so that the net
gravitational force is zero?
My confusion lies with how to solve for the position of the third planet. I am given this equation to find the force between two planets at a given distance $r$:
$$F = {GMm}/{r^2}$$
With this I can then set the sum of the two forces to zero:
$$0 = {GMm_3}/r_1^2 + {Gmm_3}/r_2^2$$
My confusion lies in the fact that both $r_1$ and $r_2$ are unknown. However, we do know that $r_1 + r_2 = d$. But I am confused with how to solve for either $r_1$ or $r_2$.
| Check the signs in your equation. Draw the three planets in a line with the central planet at the origin. What is the direction of the force each of the side planets exert on the middle one?
If you resolve the above problem you can approach solving the equations. Notice that you have as many equations as you have unknows. Therefore each equation lets you (at least in principle) express one unknown as the function of the others which reduces the problem to one with smaller number of unknows (although possibly with a more complicated equations).
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Is the earth expanding? I recently saw this video on youtube:
http://www.youtube.com/watch?v=oJfBSc6e7QQ
and I don't know what to make of it. It seems as if the theory has enough evidence to be correct but where would all the water have appeared from? Would that much water have appeared over 60 million years? Also what would cause it to expand. The video suggests that since the time of dinosaurs the earths size has doubled in volume, how much of this is and can be true?
[could someone please tag this, I don't know what category this should come under]
| There are several indicators that the Earth is both Growing ( increasing Mass ),
and Expanding ( increasing volume with or without mass increase ).
1) The continents and the submerged continental fragments fit together
on all the sides simultaneously at a radius of between 53.88 % R of the current
radius ( excluding the fragments ), up to about 58.5% R ( including all the
submerged fragments ). This simultaneous adjoining of all the continents
occurred up through the end of the Permian, when one of more bad events
started the next wave of growth and expansion. The end of the Permian
was associated with a 120 meter world wide drop in sea levels. The Reefs
that formed during the Permian are now 4,000 to 8,000 feet or more above the
current sea levels. Some other older Reefs are at still higher elevations,
like the Reef in the middle of the Salar de Uyuni in Bolivia. ( Unified Salt ).
The Dinosaurs were not physically possible in the current surface gravity
of the Earth, but they had no problem in the surface gravity of the times
that they lived in. You might note that Dinosaur sizes, and later on,
Mammal sizes diminished with the passage of time. The Math Theory here is
that the surface gravity of the Earth was/is inversely proportional to the
cube root of the dinosaur mass expressed as equivalent Male African
Elephant volumes. So when the biggest dinosaurs reduced in size to
be only 8 Elephant Volumes, then the cube root of 8 = 2, and inverting
gives a value of 1/2 g ( now ). The trouble is the When ???? How many
Million years ago did this occur? The maximum size should be less than
13 Elephant volumes for the largest sized dinosaur with a lowest gravity
being greater than 0.425 g ( now ).
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Creation of matter in the Big Bang I appreciate your patience to my neophyte question. I am working on my dissertation in philosophy (which has nothing or little to do with physics) about the "problem of naming." Briefly what I am arguing is that when we name something, we stop it from being anything or everything else. It is a phenomenological question and has a lot to do with language as an object.
My question for you is that, is it true that all matter was somehow formed in the Big Bang or in those famous three minutes following? I think I understand that helium and hydrogen were formed and are they then to be considered the basis of all matter today? A friend said to me a long time ago that we are made of the same atoms that were present at the Big Bang; could this possibly be true? (And how wonderful if it is...)
| Without there being space or time before the big bang, the laws of physics wouldn't work. This means that everything possible AND impossible could and would happen in that instance. Without the conservation of matter in effect, matter was probably created at the big bang.
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At what speed does our universe expand? Conceivably it expands with the speed of light. I do not know, but curious, if there is an answer. At what velocity, does our universe expand?
| The recessional velocity is given by Hubble's Law and it's proportional to the proper distance from the observer to the object (galaxy or other distant object):
$$v=H_0 D$$
where $H_0$ is Hubble's constant.
As you can see, the recessional velocity beyond a certain distance, known as the
Hubble distance, can be much greater than the speed of light (in vacuum). This does not violate the relativity because the recession is not a movement through the space, it's the expansion of the space itself, between the objects.
For more details check the Wikipedia article: http://en.wikipedia.org/wiki/Hubble%27s_law
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In dimensional analysis, why the dimensionless constant is usually of order 1? Usually in all discussions and arguments of scaling or solving problems using dimensional analysis, the dimensionless constant is indeterminate but it is usually assumed that it is of order 1.
*
*What does "of order 1" mean? 0.1-10?
*Is there any way, qualitative or quantitative, to see why the dimensionless constant is of order 1?
*Are there exceptions to that? I mean cases where the dimensionless constant is very far from 1? Could you give some examples? Can such exceptions be figured out from dimensional analysis alone?
| The answer follows from considering why we use dimensional quantities in the first place. So, given that the laws of physics can be formulated in a dimensionless way, how come we've ended up with using dimensional quantities. The answer is that physical quantities that we can readily measure at the macroscopic scale have certain scaling properties, and these scaling properties are not going to be the same for quantities. This means that if a simple relation exists between two quantities at the fundamental level, the corresponding relation between the rescaled quantities expressed in the units that macroscopic observers would express their experimentally accessible results in, may involve extremely small or large constants, and would therefore not be discovered early on by the first few generations of physicists. The missing relations make it necessary to assign incompatible dimensions to the physical quantities
When the missing relations are later found, they appear with dimensional constants that when expressed in the conventional units, will be extremely large or small. Compared to these typically extremely large numbers, the values of dimensionless constants will typically be much smaller. If extremely large or small dimensionless constants still appear, you can often frame that in terms of some other scaling effect. So, just like the macroscopic World we live in is approximately an infinite scaling limit that you could just as well describe using dimensionless scaling parameters (just start by using natural units $c=\hbar = G = 1$ and then replace these constants by dimensionless scaling parameters), some other scaling limit will involve some other dimensionless constant becoming very large or small.
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Why does the weighing balance restore when tilted and released I'm talking about a Weighing Balance shown in the figure:
Press & Hold on onside of the horizontal beam and then release it. It makes some oscillations and comes back to equilibrium like shown in the figure.
Both the pans are of equal equal masses. When the horizontal beam is tilted by an angle using external force, the torque due to these pan weights are equal in magnitude & opposite in direction. Then why does it come back to it position? What's making it to come back?
| If it would only be the weights exerting torque, the balance would be in equilibrium at all angles. What makes the balance go back to the horizontal position is the fact, that the center of mass is below the beam. consider this picture
The needle exerts a torque too, so you have more torque on the side, where the plate is higher. You can have more subtle configurations (like in your picture, where the beam is rounded below) but the mechanism is the same.
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Must the action be a Lorentz scalar? Page 580, Chapter 12 in Jackson's 3rd edition text carries the statement:
From the first postulate of special relativity the action integral must be a Lorentz scalar because the equations of motion are determined by the extemum condition, $\delta A = 0$
Certainly the extremeum condition must be an invariant for the equation of motion between $t_1$ and $t_2$, whereas I don't see how the action integral must be a Lorentz scalar. Using basic classical mechanics as a guide, the action for a free particle isn't a Galilean scalar but still gives the correct equations of motion.
| The key point made by Jackson here is that the Lagrangian is physically significant since it determines the equations of motion; via the extremum condition as it so happens.
Firstly, the action integral is taken along a path which is a Lorentz invariant. Secondly, since the Lagrangian is physically significant, then it should also map the same domain to the same real number from the first postulate of Special Relativity. It then follows that the action integral has to be a Lorentz scalar.
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How can two different 12V batteries have different amperage for same resistance? My local car parts dealer presented me his inventory of car batteries. One 12V battery had a 'cold crank amperage' of 600amps. The other 12V battery had a 'cold crank amperage' of 585amps. Questions:[1] If the resistance, my truck, is constant for both batteries, how can the amperage ratings differ? [2] What is does a battery's 'cold crank amperage' really mean?
| It is due to non zero internal resistances being different. If you make a short circuit, the current won't be infinity but of a finite value. It is the internal resistance who limits high currents.
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GUT that includes all 3 particle families into a large group? Explaining SU(5) GUTs (using the first particle family as an example) in the last SUSY lecture 10, Lenny Susskind mentioned that there are at present no ideas how to combine simultaneously all 3 particle families into a large GUT theory.
I somehow dont believe him, suspecting that he just didnt want to talk about this :-P...
So, are there any ideas around how to incorporate all 3 families into a larger structer?
If so, I would appreciate explanations about how it works at a "Demystified" level :-)
| The problem of families in GUT is sometimes referred as an "Horizontal symmetry". There are two lines of work, roughly: those which get a continous symmetry, say SU(3), and then all the gauge malabars, and those which add a discrete symmetry, such as A4. Of course in both cases, a serious GUT should show everything embodied in a larger simple group. E8 has some value because it can go down to E6xSU(3), and E6 can lodge chiral fermions (but then perhaps this SU(3) does not work as it should, in more detailed examination) Other alternatives are just growing up SO(2n) until everything fits... You always have V+A currents you dont want, plus a bag of any of the usual problems in phenomenology.
Zee is the adecuate source to check if you want to look deeper in this topic.
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Introducing emf of a chemical cell as a hint towards quantum mechanics Today I had a discussion with a colleague who teaches electricity and magnetism to 2nd year undergraduate physics students. He is seeking the best way to explain how is the emf generated inside a battery with a minimal appeal to physics beyond classical. We have lamented that some textbooks refer to "non-electromagnetic chemical forces" since all of chemistry is essentially electrostatics+quantum mechanics.
Our proposal is to draw the students' attention to the existence of atoms which can not be explained by classical mechanics + classical electromagnetism. In the same vein the forces on charge carriers in galvanic cells are electromagnetic but the response is not classical. Thus a battery "amplifies" non-classicality to the level macroscopic electricity. It is not that forces are non electrostatic but the systems response is not classical. Can you recommend textbooks or online sources that use/expand this idea?
(the main tag for this question should be "teaching" but I'm too much of a rookie here to create one)
| Each chemical species has a specific strength of oxidation and/or reduction. The Redox potential is what chemists use to calculate the electrode potential of a battery:
E°cell = E°red(cathode) – E°red(anode) = E°red(cathode) + E°oxi(anode)
Wikipedia has a nice compilation of Redox potentials, which are used in the above formula to calculate the voltage produced by a electrochemical cell.
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Why did my liquid soda freeze once I pulled it out of the fridge and opened it? This isn't a duplicate to "Why did my liquid soda freeze once I pulled it out of the fridge?". My question is why soda froze after it was opened. Opening a can or bottle seems to have a larger effect than just jostling it.
Is it because of the disturbance noted in the previous question? Is it related to the pressure decrease? Is it because of the release of some CO2 when it was opened?
| It is because of the Ideal Gas Law, PV=nRT. The soda is in equilibrium inside the sealed bottle or can. But when you open it, that equilibrium is disturbed. The pressure is decreased so the temperature will also decrease.
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Physical interpretation of describing mass in units of length I'm working in Taylor and Wheeler's "Exploring Black Holes" and on p.2-14 they use two honorary constants: Newton's constant divided by the speed of light squared e.g. $G/c^2$ as a term to convert mass measured in $kg$ to distance.
Without doing the arithmetic here, the "length" of the Earth is 0.444 cm; and of the sun is 1.477 km. To what do these distances correspond? What is their physical significance, generally?
| I'm not sure it's terribly helpful, but it seems like the following analysis helps explain dmckee's response.
The force of gravity is
$F = G \frac{m M}{r^2}$.
Rearranging and dividing by $c^2$ gives
$\frac{G}{c^2} = \frac{F r^2}{M (m c^2)}$
where the $mc^2$ is the rest mass energy $(E_0)$ of the object experiencing the force caused by mass $M$. When you multiply through by the mass of the "large" object you get
$M \frac{G}{c^2} = l = \frac{F_g r^2}{E_0}$
Since we are interested in the length $l$, at that distance we have
$M\frac{G}{c^2} = r = \frac{F_g r^2}{E_0}$
or simply
$E_0 = F_g r$.
In words, this is the distance at which the energy of the system due to the rest mass of an object in a gravitational field is the same as the potential energy due to gravitation.
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How can it be that the beginning universe had a high temperature and a low entropy at the same time? The Big Bang theory assumes that our universe started from a very/infinitely dense and extremely/infinitely hot state. But on the other side, it is often claimed that our universe must have been started in a state with very low or even zero entropy.
Now the third law of thermodynamic states that if the entropy of a system approaches a minimum, it's temperature approaches absolut zero.
So how can it be that the beginning universe had a high temperature and a low entropy at the same time? Wouldn't such a state be in contradiction to the third law of thermodynamics?
| Entropy is not the existence of heat or energy, but is more accurately described as the spread of energy. A universe with high heat and low matter density has very low entropy, the same way that a cup of hot water has low energy distribution when compared to a cold pool. If you throw the hot water into the cold pool the heat will spread throughout the pool as would be expected by the laws of thermodynamics, similarly the matter and energy from the big bang spread throughout the universe from a single point of low entropy.
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Can superconducting magnets fly (or repel the earth's core)? If a superconducting magnet and appropriate power supply had just enough $I\cdot s$ (current $\cdot$ length) so that when it was perpendicular to the earth's magnetic field, the force of the interaction was just enough to excede the force exerted on the object from gravity. And it was rotating so the angular momentum of the vehicle was just high enough so it wouldn't flip over, would the vehicle fly?
Assuming the vehicle is a 1000 kg (and the earth's magnetic field is $0.3$ gauss) I calculated that with $6.54\cdot10^8$ meter amperes you just about reverse the force on the vehicle.
Now assuming a $100$ meter diameter, that leaves $6.54 \cdot 10^6$ A, which is less then the current in a railgun, but still a lot.
The problem is that the force normal is no longer so normal. It will want to flip the vehicle so the magnet is the other way. Now we would need to spin the vehicle fast enough, so that it has rotated 180 degrees faster then it would take for the force of the magnet to flip the vehicle 180 degrees. How would you go about calculating this part?
| The bigger problem would be keeping it cool and not going over the magnetic threshold of the material. You really wouldn't need to worry about the spinning since your using the flux trapping property of the super conductors. Essentially once you produced the field needed you could trap it in a conducting loop, and as long as it was cooled properly you would have almost a perfect field for decades.
| {
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under what conditions happen the anti-Zeno effect? As you might know, the Zeno effect is intuitively expressed as what happens when a system is measured in intervals smaller than the half life of the state it is currently on. As a consequence, the state has a negligible probability of doing a transition and is kept 'stuck' in its current state, making the effective evolution operator the identity.
I don't know an equivalent picture for the Anti-Zeno effect. Under what conditions does it happen and why? does the above picture is merely interpretational or is fully accurate at a fundamental level?
| The Anti-Zeno effect is when you have a transition from a state enhanced by a decoherence source. I can show you a toy model where it happens:
consider four states A,B,E and Z. Z is the ground state, and A and B are two nearly degenerate excited states and E is a super-excited. B has no matrix element to Z and A has matrix elements to Z. Then start the system in A, and keep measuring if you are in B by shining a laser tuned to the B-E transition. The system will, through the ordinary Zeno effect, be stuck in A, and then will decay to Z through the A->Z transition.
If you turn off the laser, the system will oscillate between A and B, and will only decay in the A portion.
| {
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Why does current alternate in an AC generator? I understand how generators work, but I can't for the life of me conceptualize why the current in an AC generator reverses every 180 degrees!!! I understand that, using the right hand rule, it can be seen that the current flows around the coil in a generator in one direction, but I can't see why the current reverses after half a rotation!
I've been looking at the animations on this page to try and figure it out, but I just can't get there.
In addition, I don't understand the concept of how split ring/slip rings work? I know split ring is for direct current, but not really why. For instance, if asked how could I 'explain the difference between a split ring or slip ring'?
| Set up a magnet around a coil, such that the Magnet has a field that is constant in magnitude, and has vector form ${\vec B} = B_{0}(\cos (\omega t) {\hat z} + \sin (\omega t) {\hat x})$. Orient the coil so that it lies in the $x-y$ plane and thus has a normal that points in the $z$ direction. If the overlap of the coil's area and the magnetic field is $A$, then the net flux through the coil as a function of time is given by $\Phi=B_{0}A\cos\theta$. Then, Faraday's Law ($V_{ind}=-\frac{d\Phi}{dt}$) tells us that the induced voltage through the loop is given by $V_{ind}=B_{0}A \omega \sin \omega t$, which reverses every half cycle.
By going back to the original inducing magnetic field, you can see that this reverses every half cycle because the direction of the magnetic field also reverses every half cycle.
| {
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What determines the apparent radius of the rainbow? Let's say I know how to compute the apparent radius of a rainbow from the viewpoint of the observer: take a photo of the scene, measure the distance to a known reference object, and its dimensions. Using triangle similarity, I can extrapolate the radius of the rainbow.
But my question is: which physical phenomenon determines the radius?
| It depends on where the sun is. If it is near the horizon (behind you) and in front of you there are water droplets, then you will see a rainbow with a radius (in angular measure) of about 42 degrees, because each water droplet returns a cone of light, whose axis is parallel to the direction to the sun and whose aperture is roughly $2 \cdot 42 = 84$ degrees.
I've never seen better explanations of dozens of phenomena concerning rainbows than in Walter Lewin's lectures.
| {
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Experimental evidence for parallel universes/multiverses My idea of physics is that it is a collection of mathematical laws relating observables. And that one can perform alot of mathematical derivations on these laws to produce new laws between observables. My question is how does one translate a mathematical equation into 'there exist other universes like ours'?
How does one derive that there exist other universes, what phenomena do they explain?
Which observables suggest other universes?
| There are speculative theories that suggest areas of cold/hot patches in the cosmic microwave background are 'bruises' caused by collisions of expanding bubble universe... These are not 'parallel' in your sense though, more they are just different regions of space-time.
| {
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Why Do Hurricane Balls Spin So Fast? I was wondering if anyone could offer an explanation as to why the balls described in this video spin so fast.
Here's the setup: Two metal balls are wielded together. When spun with air, they acquire a massive amount of rpm.
| The balls never spin faster than the velocity of the air being blown on them. Notice that when the presenter blows on the hurricane balls, he angles the mirror towards himself. He does this so the balls aren't blown off the mirror (he's blowing pretty hard). Also, it seems they are spinning ridiculously fast due to the frame rate of the video.
Edit: When I referred to the velocity of the balls, I was referring to their tangential velocity.
| {
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Could the Schrödinger equation be nonlinear? Is there any specific reasons why so few consider the possibility that there might be something underlying the Schrödinger equation which is nonlinear? For instance, can't quantum gravity (QG) be nonlinear like general relativity (GR)?
| @Ron Maimon has given the canonical answer to this: the wavefunction is probabilities, and to preserve probabilities one must have a linear equation (indeed, also a norm-preserving evolution operator).
I offer another viewpoint, in the style of how Einstein thought about relativity, i.e. two postulates. The postulate is that it is not possible to solve NP-complete problems in polynomial time. Abraham and Lloyd showed that if quantum mechanics were non-linear at all, then this would be possible.
Aaronson has a nice paper, the start of which references a large literature on why quantum mechanics has to be the way it is.
| {
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Since the universe is expanding, does anything ever occupy the same point in space? Let's say we can observe expansion in a supercluster.
We define origin of our frame of reference at the center of the supercluster.
We observe an object/atom at point A at time T. The object is motionless relative to the origin.
We wait for expansion until T+ΔT and again observe the object.
Is the object at A or somewhere else?
| This seems to be dependent on the manner in which the super-cluster expands.
If expansion, mass, and energies expand perfectly synchronous relative to each other and the point defined as "center", then the "center" remains relative to all other points. We know this not to be the case, therefore, the "center"will constantly relocate in relation to all other points of the super-cluster as defined by the mechanisms affecting all other points.
Where's the center of the vortex in the toilet bowl? Constantly in flux due to motions of mass relative to it.
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Acceleration vector - deceleration vs direction If acceleration of something $= - 10 \text{ m s}^{-2}$
And forwards is define as north.
Does that mean the object is getting slower (decelerating) or accelerating in the reverse direction (south)
How can you tell the difference?
|
Does that mean the object is getting slower (decelerating) or accelerating in the reverse direction (south)
It really doesn't matter. Basic kinematic formulas are designed to work just as well in either case, which is why physicists don't generally use the word "decelerating." It's just another kind of acceleration.
That being said, if you want to determine whether the object's speed is increasing or decreasing (which correspond to the popular meanings of "accelerating" and "decelerating" respectively), you can just look at the orientation of the acceleration with respect to the velocity. If the acceleration is parallel to the velocity, the object will be speeding up. If it's antiparallel, the object will be slowing down. You can see this mathematically by taking the derivative of the kinetic energy:
$$\frac{\mathrm{d}}{\mathrm{d}t}\biggl[\frac{1}{2}mv^2\biggr] = m\vec{v}\cdot\frac{\mathrm{d}\vec{v}}{\mathrm{d}t} = m\vec{v}\cdot\vec{a}$$
So the sign of the dot product $\vec{v}\cdot\vec{a}$ tells you whether the speed is increasing or decreasing.
Do note that velocity is reference frame-dependent. So two different inertial observers looking at the same object at the same time could have differing conclusions as to whether it is speeding up or slowing down. That's one big reason why the distinction is not important in physics.
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How long does a permanent magnet remain a magnet? I have a bunch of magnets (one of those game-board thingies) given to me when I was a school-going lad over 20 years ago, and the magnets feel just as strong as it was the day it was given.
As a corollary to this question Do magnets lose their magnetism?, is there a way to determine how long a permanent magnet will remain a magnet?
Addendum:
Would two magnets remain a magnet for a shorter duration if they were glued in close proximity with like poles facing each other?
| If a permanent magnet could "decay" at the rate given in Rook's answer above there would be none found in geological strata.
A permanent magnet has a permanent orientation of the magnetic moments in a specific vectorially additive direction depending on small crystal domains. To change, i.e. be demagnetized, the magnetic moments have to be randomized by either an external magnetic field or excess heat/melting or vibrations possibly. If nothing like that happens it should be stable. Little magnets in a box left undisturbed would not change magnetisation unless a random magnetic field was in the area .
Non magnetic iron left undisturbed will acquire a field from the magnetic field of the earth, so some change in the orientation of the field could happen to these little magnets, depending on how they lay with respect to the weak field of the earth.
| {
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How can one localize the massless fermions in Dirac materials? I noticed that finite electric potential cannot localize the low energy excitations in a graphene sheet. Is it possible to localize the massless fermions in the surface band of topological insulators with a magnetic field?
I found a paper dealing with a similar problem: http://apl.aip.org/resource/1/applab/v98/i16/p162101_s1
| This issue is a well known problem in high energy physics which is called " Neutrino Billiards". You can find a full description about it in:
Ref:
Berry, M.V. and R.J. Mondragon, Neutrino Billiards: Time-Reversal Symmetry-Breaking Without Magnetic Fields. Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences, 1987. 412(1842): p. 53-74.
And in the case of graphene C.W.J. Beenakker have a good paper about this problem:
Physical Review Letters, 2006. 96(24): p. 246802. or it's Arxiv:0603315 counterpart.
in summary: for making bound states from massless Dirac fermions you must use a mass term in each bound instead of electric potentials. So the probability of Klein tunneling set to zero and fermions became confined.
| {
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Why are color values stored as Red, Green, Blue? I learned in elementary school that you could get green by mixing blue with yellow.
However with LEDs, TFTs, etc. you always have RGB (red, green, blue) values?
Why is that? From what you learned in elementary yellow would be the 'natural' choice instead of green.
| The inherent difference is things that emit colors, e.g. LEDs, and things that place substances on a surfaces to color them, e.g. printers. In the latter case, the CMYK (cyan, magenta, yellow, black) space is commonly used, rather than RGB, so you were on the right track with yellow. BTW, black is there because the black produced by mixing C, M, and Y doesn't look as good and printers generally print black stuff on white paper).
On a philosophical note: color is not a physical property of an object, i.e. we should not really talk about a "blue car"; it is an example of conscious experiences known as qualia.
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Is there some explanation for $y_t=1$ The Yukawa coupling of the top quark is Dirac-natural in a too excellent way, it is within one sigma experimentally, and within 99.5% in absolute value, of being equal to one. Without some symmetry, it seems too much for a quantity that is supposed to come down from GUT/Planck scale via the renormalization group. Is there some explanation for this?
| This is a very naive answer or, in fact, it is not an answer. Among all numbers of order one, is not $y_t=1$ the most likely value, i.e., the statically expected value? Why do we need an explanation for $y_t=0.995$ and not for, say, $y_t=0.629$?
| {
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Superluminal neutrinos
I was quite surprised to read this all over the news today:
Elusive, nearly massive subatomic particles called neutrinos appear to travel just faster than light, a team of physicists in Europe reports. If so, the observation would wreck Einstein's theory of special relativity, which demands that nothing can travel faster than light.
—source
Apparently a CERN/Gran Sasso team measured a faster-than-light speed for neutrinos.
*
*Is this even remotely possible?
*If so, would it be a real violation of Lorentz invariance or an "almost, but not quite" effect?
The paper is on arXiv; a webcast is/was planned here.
News conference video here
| Suppose this is real, that the neutrinos arrive very slightly faster than light would through the vacuum. Wouldn't that point to there being a slightly higher c which actually limits speeds, and some slight slow-down for light from this maximum due to interactions of the electromagnetic field with other particles, including virtual particles?
After all, you can move an electron faster than a photon in glass, and we don't call it the end of relativity, we call it Cherenkov radiation.
So the definition of refractive index might need adjusting, but effectively the vacuum has a non-zero refractive index, or rather the vacuum is not entirely empty. Which we know.
It makes sense that a neutrino is not subject to the same interactions, given its famed reluctance to interact with anything. Perhaps it is just an indication that the particles in a vacuum are more likely to be electromagnetic-interacting than weak-interacting.
Or am I labouring under a false premise? Is the speed of light in a vacuum already adjusted for virtual particle interactions?
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What is the "shape" of atomic/subatomic particles? I apologize in advance for my ignorance if this is a question with an obvious answer... I am not experienced in this field. But are such particles in the universe points with a charge, or are they very small spheres with a charge? Or does it not even matter in the end? This isn't homework, it's just curiosity.
| continuing with @lusken 's answer, the atom is perceived as a fuzzy ball with a highly dense nucleus (mainly point size, compared to the size of the atom itself) and the fuzzy boundary because of the electron cloud.
The electron cloud themselves appear in different probability distribution, which gives different "shapes" to them.
EDIT1: where s is for electron with 0, p for spin 1 and so on d = 2 and f = 3. and each s,p,d,f have different suborbitals which are depicted as in the figure as
*
*$s$
*$p_x$, $p_y$, $p_z$
*$d_{xy}$, $d_{yz}$, $d_{zx}$, $d_{x^2-y^2}$, $d_{z^2}$
| {
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How do you find the uncertainty of a weighted average? The following is taken from a practice GRE question:
Two experimental techniques determine the
mass of an object to be $11\pm 1\, \mathrm{kg}$ and $10\pm 2\, \mathrm{kg}$.
These two measurements can be combined to
give a weighted average. What is the uncertainty of the weighted average?
What's the correct procedure to find the uncertainty of the average?
I know what the correct answer is (because of the answer key), but I do not know how to obtain this answer.
| I agree with @Ron Maimon that these ETS questions are problematic. But this is (i think) the reasoning they go with. Unlike @Mike's assumption you should not take the normal average, but as stated in the question the weighted average. A weighted average assigns to each measurement $x_i$ a weight $w_i$ and the average is then
$$\frac{\sum_iw_ix_i}{\sum_i w_i}$$
Now the question is what weights should one take? A reasonable ansatz is to weigh the measurements with better precision more than the ones with lower precision. There are a million ways to do this, but out of those one could give the following weights:
$$w_i = \frac{1}{(\Delta x_i)^2},$$
which corresponds to the inverse of the variance.
So plugging this in, we'll have
$$c = \frac{1\cdot a+\frac{1}{4}\cdot b}{1+\frac{1}{4}}= \frac{4a+b}{5}$$
Thus,
$$\Delta c = \sqrt{\left(\frac{\partial c}{\partial a}\Delta a\right)^2+\left(\frac{\partial c}{\partial b}\Delta b\right)^2}$$
$$\Delta c = \sqrt{\left(\frac{4}{5}1\right)^2+\left(\frac{1}{5}2\right)^2}=\sqrt{\frac{16}{25}+\frac{4}{25}}=\sqrt{\frac{20}{25}}=\sqrt{\frac{4}{5}}=\frac{2}{\sqrt5}$$
which is the answer given in the answer key.
Why $w_i=1/\sigma_i^2$
The truth is, that this choice is not completely arbitrary. It is the value for the mean that maximizes the likelihood (the Maximum Likelihood estimator).
$$P(\{x_i\})=\prod f(x_i|\mu,\sigma_i)=\prod\frac{1}{\sqrt{2\pi\sigma_i}}\exp\left(-\frac{1}{2}\frac{\left(x_i-\mu\right)^2}{\sigma_i^2}\right)$$.
This expression maximizes, when the exponent is maximal, i.e. the first derivative wrt $\mu$ should vanish:
$$\frac{\partial}{\partial\mu}\sum_i\left(-\frac{1}{2}\frac{\left(x_i-\mu\right)^2}{\sigma_i^2}\right) = \sum_i\frac{\left(x_i-\mu\right)}{\sigma_i^2} = 0 $$
Thus, $$\mu = \frac{\sum_i x_i/\sigma_i^2}{\sum_i 1/\sigma_i^2} = \frac{\sum_iw_ix_i}{\sum_i w_i}$$ with $w_i = 1/\sigma_i^2$
| {
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Is there a mathematical way to describe how a flame flickers? I love the way candles flicker. It's a great effect and I almost want to see it replicated in an actual lightbulb. I was curious if there is any way to express that mathematically? I'm not that familiar with physics but I go in assuming that most things can be expressed in math, just very complex in some cases!
| If you want to determine physical causes of flame shape evolution, you would have to consider all processes taking place (chemical, phase changes, fluid dynamics, heat, ...); they are described using known differential equations, i.e. in one point in space. The exact global solution then requires those equations to hold simultaneously (coupling between processes) in all points of your domain, but such (analytical) solution it is only obtainable in very special cases. Approximate global solution is found numerically, by discretizing your problem in space and time, and requiring those differential equations to hold only in finite number of points in some approximate (though mathematically precisely defined) manner.
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understanding time: Is time simply the rate change? Is time simply the rate of change?
If this is the case and time was created during the big bang would it be the case that the closer you get to the start of the big bang the "slower" things change until you essentially approach a static, unchanging entity at the beginning of creation?
Also, to put this definition in relation to Einstein's conclusions that "observers in motion relative to one another will measure different elapsed times for the same event." :
Wouldn't it be the case that saying the difference in elapsed time is the same as saying the difference in the rate of change.
With this definition there is no point in describing the "flow" of time or the "direction" of time because time doesn't move forward but rather things simply change according to the laws of physics.
Edit: Adding clarification based on @neil's comments:
The beginning of the big bang would be very busy, but if time was then created if you go back to the very beginning it seems there is no time and there is only a static environment.
So it seems to me that saying time has a direction makes no sense. There is no direction in which time flows. There is no time; unless time is defined as change.
So we have our three dimensional objects: and then we have those objects interact. The interaction is what we experience as time. Is this correct or is time more complicated than this?
| Time is what is measured by clocks.
But how is time modelled in physical theories ?
In the Schrödinger equation time enters as an external parameter. How does this parameter correspond to the time measured by clocks ?
The following reference might be a good introduction to this and related questions concerning time and quantum mechanics : http://www.physedu.in/uploads/publication/1/7/28-1-3-The-challenging-concept-of-time-in-quantum-mechanics1401.pdf
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What is the difference between electric potential, electrostatic potential, potential difference (PD), voltage and electromotive force (EMF)? This is a confused part ever since I started learning electricity. What is the difference between electric potential, electrostatic potential, potential difference (PD), voltage and electromotive force (EMF)? All of them have the same SI unit of Volt, right? I would appreciate an answer.
| EDIT: Put simply, potential difference is the work done by electrostatic force on a unit charge, while EMF is the work done by anything other than electrostatic force on a unit charge.
I don't like the term "voltage". It seems to mean anything measured in volts. I'd rather say electric potential and electromotive force.
And the two are fundamentally different.
Electrostatic field is conservative, that is, over any loop $l$ we have $\oint_l \vec{E}\cdot\mathrm{d}\vec{l}=0$. In other words, the line integral of electrostatic field does not depend on the path, but only on end points. So we can define point by point a scalar value electrostatic potential $\varphi$, such that
$$\varphi_A-\varphi_B=\int_A^B \vec{E}\cdot\mathrm{d}\vec{l},$$
or
$$q \left( \varphi_A-\varphi_B \right)=\int_A^B q\vec{E}\cdot\mathrm{d}\vec{l},$$
so $q\Delta\varphi$ equals the work done by electrostatic force.
In pratical application, electrons (and other carriers) flow in circuits. Since electrostatic field is conservative, it alone cannot move electrons in circles; it can only move them from lower potential to higher potential. You need another kind of force to move them from higher potential to lower ones in order to complete a cycle. This other force could be chemical, magnetic or even electric (vortex electric field, different from electrostatic field), and their equivalent contribution is called electromotive force.
$$\mathrm{E.M.F.}=\int_\text{Circuit} \frac{\vec{F}}{q}\cdot\mathrm{d}\vec{l}$$
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Numerical software to manipulate a light beam in its plane wave representation? Any light field can be expressed as a sum of plane waves. Such an ensemble of plane waves is called the plane wave spectrum of the light field. The plane wave spectrum is the Fourier transform of the light field in the real space representation.
Since, this is such a basic technique, I wonder whether there is a standard tool, preferably open-source, with a simple interface via Python, Matlab or similar, to do the following (numerically):
*
*Start with a light field in real space, say a monochromatic *-Gaussian beam.
*compute the Fourier transform
*apply some operation that changes the plane wave spectrum, say a boundary between two media
*compute the inverse Fourier transfrom
*produce some output, e.g. a plot
Computationally, all it takes is a library to perform the FFT. I'm looking for a framework that would wrap this along with some phyiscal concepts.
| I use pylab numpy scipy matplotlib (and matplotlib.mlab)
examples f.i. here and doc and nice ref here
or use the online integrated python environment sagenb.org (open account or download package, upload files and start working anywhere)
nice plots ...
to do digital filters explore the online book
The Scientist and Engineer's Guide to Digital Signal Processing
and explore the videos (30 lessons) "linear systems and optimization the fourier transform and its applications" from stanford.edu.
If you have a nice graphics card you can install OPENCL, pyOpenCL, pyfft (easyer in linux box)
and you can accelerate the fft and all array operations.
EDIT add
SCIAO -- :
SciAO is an open source, cross-platform, and user-friendly toolbox
based on the Scilab/Scicos environment for modeling and simulation of
wave optics, especially the adaptive optics system
| {
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Open quantum systems and measuring devices The Copenhagen interpretation by Niels Bohr insists that quantum systems do not exist independently of the measuring apparatus but only comes into being by the process of measurement itself. It is only through the apparatus that anything can be said about the system. By necessity, the apparatus has to be outside the system. An open quantum system. Can quantum mechanics be applied to closed systems where the measuring apparatus is itself part of the system? Can a measuring apparatus measure itself and bring itself into existence?
| ''that quantum systems do not exist independently of the measuring apparatus but only comes into being by the process of measurement itself'' is a gross distortion of the Copenhagen interpretation. The latter only asserts that the particular value of measuring quantum variables of a system that exists objectively (otherwise how could it be measured) is predictable only within its intrinsic uncertainty.
The solar system is a quantum system whose state we know reasonably well in a coarse approximation appropriate to such big systems, as we know its thermal properties and quantum gravity effects play no role. All our experiments so far have been performed within this quantum system, and all our measuring instruments are part of it.
Every individual measurement we do is in fact a measurement of the state of a tiny subsystem, sometimes (spin or polarization measurement) of only a single quantum degree of freedom, and thus reveals a tiny little bit more about the state of the solar system, namely about the substate obtained by tracing out all other degrees of freedom. This tracing out is the source of decoherence, which is frequently well approximated by the Copenhagen collapse postulate.
Thus there is not the slightest trace of the mystery the OP seems to suggest.
| {
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What ring weave disposition should be the most resistant against stabbing and/or how to determine it In a website that I am studying so I can build a Chain Mail, I have found a page featuring a lot of different Ring Weaves to build them.
I want to determine which one is the best to provide resistance against stabbing strikes, so I wish to know if there are some calculations based on wire resistance, thickness and ring size that could give me a theoretical comparation between configurations.
Currently my favourite one is Japanese 6 in 1.
But I think that one like Dragonscale) has to be very resistant too.
How to know it? What calculations can I do to get at least an idea on which to prefer for this? Any help is appreciated an I am sorry for being such a n00b if my question is not appropriate for this place.
| I will hand wave on this:
The best resistance against stabbing must be metal plate, i.e. no holes where a knife can enter.
The weave is the result of trying to introduce flexibility so that a dress can be made out of the material that would allow agile motion.
The best weave is the weave that when projected against the possible penetration directions leave no projective holes, i.e the knife will hit a wire at some level, and thus it will give the same protection as sheet metal.
I would choose something like "Japanese 8 in 2", or" byzantine variant 1" if the interweaving left no extra holes. i.e. something with many inter leaved levels of wire.
| {
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Mathematical definition of Bogomol'nyi–Prasad–Sommerfield (BPS) states What is the mathematical definition of Bogomol'nyi–Prasad–Sommerfield (BPS) states, independent of any specific physical theory.
| It's a state annihilated by $E-Q$ where $E$ is the energy, or another dynamical/isometry generator, and $Q$ is the sum of multiples of other conserved charges such that one may write
$$ E - Q = \sum_{i,j} c_{ij}\{Q_i,Q_j\} $$
i.e. the difference between energy and charges may be obtained as an anticommutator of some supercharges – Grassmann-odd symmetry generators acting on the system. It's easy to show, by looking at the vanishing expectation value of $E-Q$, that BPS states are equivalently those that are annihilated by some supercharges $Q_i$ or their combinations. Consequently, the space of states obtained by the action of all the generators on the BPS state has a smaller dimension than the dimension of the long multiplet – $2^{N_{\rm sup}/2}$. We call such multiplets short.
| {
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Quick question concerning the Heisenberg model I got a small, rather technical question concerning the Heisenberg model.
(It is technical indeed.)
Consider the Heisenberg Hamiltonian:
$H = \sum_{(i,j)} S_{i} S_{j}$ = $- \frac{J}{V} \sum_{q} \gamma_{q} S_{q} S_{-q}$.
with
$\gamma_{q} = 2 \sum_{\alpha=x,y,...}cos(q_{\alpha})$
and the Fourier transform
$S_{i}=1/V \cdot \sum_{q} S_{q} e^{iqr_{i}}$
Where we assume a lattice constant of a=1 and impose periodic boundary conditions on a hypercube of edge length L and respective volume V.
Here comes my question: I don't see how one arrives at the equation for the fourier
transformed H in detail. I mean...i see where the cosines are coming from if you
only have -q and q left and suspect that one has to apply an identity like
$\delta_{ij} = \sum_{q} e^{iq(r_{i}-r_{j}}$ but somehow it doesn't work out for me and
even though it has nothing to do with the physics it leaves a bad feeling behind.
I'd be really thankful if someone could help me.
Best regards and thanks in advance.
| The thing to keep in mind is that that formula only holds because you are coupling nearest-neighbor spins. Let's pretend we're in 1D (the generalization for higher dimensions is trivial). When you replace the operators by their fourier transformed representations that you provided, you get
$$
J\sum_i S_i \cdot S_{i+1} = \frac{J}{V^2}\sum_i \sum_{k,q}S_q e^{iqr_i} \cdot S_k e^{ikr_{i+1}}
$$
Since I'm only coupling nearest neighbors, the spatial coordinates in the exponential are always just one site away from one another, so I can write
$$
S_q e^{iqr_i} \cdot S_k e^{ikr_{i+1}} = e^{ik}(S_q \cdot S_k)e^{i(q+k)r_i}
$$
Now, you almost noted the identity you need in your comment, but what you really need is
$$
\sum_{i}e^{i(q-k)r_i}=V\delta_{qk}
$$
I hope this helps
| {
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Does the speed of sand flow in a hourglass depend on a height of a sand column above the hourglass neck? In a hourglass, does the sand flow through the neck depend on the amount of sand in the upper glass? If we consider a sand flow analogous to fluid flow, then it should depend linearly, but in that case amount of sand to represent the given time would rise squared depending on time?
| Fluid approximations do not work well at the scale of sand in hourglasses for most hourglasses. Almost all of the sand is statically braced against the walls and floor of the hourglass. Instead, you have a small region of instability above the hole where it is not possible for the sand to be braced; as that unstable portion falls through the hole, more falls in from above. There is slow flow of sand downwards, but it's still mostly statically braced. Thus, to a first approximation, flow rates are determined by grain size and shape (and material), and the immediate geometry around the hole (predominantly the size).
| {
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why making a surface "super" smooth increases the coefficient of friction? I read that:
If you take a rough surface and make it smooth, the coefficient of friction decreases. But if you make it super smooth, then the coefficient of friction increases. How come?
| What do you mean by super smooth?
I remember a note (again) by Feynmann that said if you made your test surface so clean that there is absolutely no dirt or impurities on it, then the super clean surface would actually attach to anything sliding on it, making the apparent friction coefficient higher.
Is this what you mean by super smooth?
In that case, take a copper plate. It is ideally made of copper atoms. Its surface will be dirty, filled with other molecules.
If you now imagine that you have the tool to clean it so well that just the copper atoms are on the surface and nothing else, you will actually get a very reactive - in the chemical sense - surface. "Naked" atoms will bind to anything that passes by, and if you try to make something slide over it, they will make bounds and stick very well.
| {
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Does the Opera result hint to a discrete spacetime? Could the Opera result be interpreted as some kind of hint to a discrete spacetime that is only seen for high enough energy neutrinos?
I think I've read (some time ago) something like this in a popular article where among other things tests of quantum gravity theories, that assume a discrete spacetime, are explained.
Looking around in blogs and other places in the web, I notice that this is disussed seldom or not at all... ``
| In several theories, space itself is discrete, somewhat in relation to the Planck length, $$l_p = \sqrt{\frac{\hbar G}{c^3}} \simeq 1.616199 \times 10^{-35}\quad m$$ .
More specifically in loop quantum gravity, Carlo Rovelli's 1998 overview paper states the following:
The spin-networks picture of space–time is mathematically precise and physically compelling: nodes of spin networks represent elementary grains of space, and their volume is given by a quantum number that is associated with the node in units of the elementary Planck volume $$V = \left( \frac{\hbar G}{c^3} \right)^{3/2}$$
So, from what I understand of LQG, space has always been discrete. However, mathematically, space being discrete does not imply that time also is (which would mean that spacetime is discrete). A counter example in 2D would be the floor and ceiling functions.
Concerning the OPERA results, let's keep in mind that several explanations have been published which don't allow for supralumnial neutrinos, cf this Universe Today article or this Bad Astronomy article.
I am relatively new here, and I might not have fully answered your question, so feel free to post comments or even modify my answer to improve it. Thanks!
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Orbital radius of Geo-stationary satellite Could you please tell, Why all the geo-stationary satellites are to be dropped at same height from earth? Why can't it be closer or away from its regular orbit(ie, 35,000 km)? If all satellites are dropped in the same orbit, then will not those collide one another?
| Geostationary satellites are less likely to collide since they are all moving in the same direction. But they aren't all perfectly positioned and so do drift - even 'perfectly' positioned ones do drift in a figure of 8 around their intended point.
But if there ever is a collision, or an explosion, the debris is going to stay there essentially for ever - unlike stuff in LEO which will re-enter the atmosphere. And you don't have an alternative geostationary to choose from. If there is ever a large cloud of debris in the GSO slot for N. America then you are all going to have to switch to cable.
On the other hand GSO is become less important as more and more comms switches to fibre.
| {
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What are quarks made of? So atoms are formed from protons and neutrons, which are formed from quarks.
But where do these quarks come from? What makes them?
| The standard mainstream answer is to consider them as fundamentals. Another standard, but not mainstream, answer is that we call generically "preons" to the hypothetical components of quarks and leptons. The most stablished -arguably- preon theory is Harari-Shupe, sometimes referred to as "rishon theory", but there are others.
Without preons, string theory could be also an answer but not in the line of your question; quarks and leptons would be equivalent to some string states, so not "made of", but "same as". Similarly, in Kaluza Klein theory: the quarks and leptons are expected to be special states of the compactified theory. Of course, again, this is the mainstream. Theorists have also proposed models where the states are Rishons.
The middle way, you could have the theories that propose to produce quarks and leptons out of geometry. These theories usually worry a lot about gravity.
Last, you have the non-standard theories. I myself have one of them, the sBootstrap, and no doubt that some other people will intend to answer you by proposing their favourited theory.
| {
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Know altitude and speed of an object in orbit, with true anomaly I'm quite stuck with this problem.
I know that I have an object in orbit. I know the eccentricity of that orbit, as well as the semi-major axis of the orbit.
Giving a true anomaly, how do I find the speed and altitude of that object? The true anomaly is the angle between the line made with the focus of the ellipse and the position of the object.
Thank you!
P.S. I'm more looking for a general help, more than a specific answer. That's why I didn't give any numbers.
| As far as I can tell the true anomaly is the same type of angle used in the standard solution of the Kepler problem since there we assume the sun is at a foci.
When solving the equations of motion for a Keplerian orbit we obtain $r\left(\theta\right)=\frac{a\left(1-e^{2}\right)}{1\pm e\cos\theta}$ (- if $r\left(0\right)$ is through the origin and + if it is away from the origin) and we can express the velocity as $v=\sqrt{\mu\left(\frac{2}{r}-\frac{1}{a}\right)}$ where $\mu$ is the standard gravitational parameter which in the two body case is just $G\left(m_{1}+m_{2}\right)$.
The derivation of these formulae can be found in many mechanics textbooks. For example, Taylor - Classical Mechanics - chapter 8.
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Dependence of Friction on Area Is friction really independent of area? The friction force, $f_s = \mu_s N$. The equation says that friction only depends on the normal force, which is $ N = W = mg$, and nature of sliding surface, due to $\mu_S$.
Now, less inflated tires experiences more friction compared to well inflated tire. Can someone give clear explanation, why friction does not depend on area, as the textbooks says?
| It is all about the distribution of pressure under the contact. For a block of uniform weight the pressure can be assume almost constant under the area and so when traction is broken it will happen all at once all over with a force of $\mu N$ as you stated.
But for other geometries, or for elastic parts (like tires, or marbles or billiard balls on felt) the contact pressure has various other shapes. The parts with the highest pressure (sometimes in the middle, and often at the edges) are going to stick more than the unloaded parts. The result is complicated, but in the end we call the total traction to achieve full slipping still $\mu N$, but with $\mu$ a different value that for the block above, even if the materials are the same.
A lot of scientific papers are written on the subject of how traction affects the contact properties and vice versa. Dealing with a coefficient $\mu$ for the force and not the area makes it easier to summarize the results, but in reality the pressure shape over the area is the ultimately in control here.
| {
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Are specific heat and thermal conductivity related? Are there any logical relationship between specific heat capacity and thermal conductivity ?
I was wondering about this when I was reading an article on whether to choose cast iron or aluminium vessels for kitchen.
Aluminium has more thermal conductivity and specific heat than iron ( source ).
This must mean more energy is required to raise an unit of aluminium than iron yet aluminium conducts heat better than cast iron.
Does it mean that aluminium also retains heat better ?
How does mass of the vessel affect the heat retention?
| This is in address to your last two questions:
*
*Which retains heat better?
*How does mass effect heat retention?
You introduce two material properties (mass, specific heat) that seemingly affect the heat retention but do not give you the whole picture in of themselves. However, we can combine them to give us a useful measure of heat retention; this is also known by many other names (thermal mass, volumetric heat capacity, thermal capacitance).
Thermal Mass
$C_{th} = m C_p $
| {
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Inverted Harmonic oscillator what are the energies of the inverted Harmonic oscillator?
$$ H=p^{2}-\omega^{2}x^{2} $$
since the eigenfunctions of this operator do not belong to any $ L^{2}(R)$ space I believe that the spectrum will be continuous, anyway in spite of the inverted oscillator having a continuum spectrum are there discrete 'gaps' inside it?
Also if I use the notation of 'complex frequency' the energies of this operator should be
$$ E_{n}= \hbar (n+1/2)i\omega $$
by analytic continuation of the frequency to imaginary values.
| The wave functions that are not $L^2$-integrable play no direct physical role. You may get such "mathematically nice" functions e.g. by the analytical continuation from the stable (non-inverted) harmonic oscillator but they won't have the same interpretation. That's easy to see: as you noticed, the analytic continuation gives you imaginary energies which can't be the eigenvalues of a Hermitian operator.
The actual eigenvalues are arbitrary real numbers (the energy may always be made higher positive, by the kinetic energy, as well as more negative, by the unbounded-from-below potential) and I am convinced that each of them has a degeneracy of two, one wave moving right and one moving left in some convention. There are not even "exceptional gaps" where the degeneracy would change.
The formal solutions with $E_n=\hbar(n+1/2) iw$ still exist as poles in the transition amplitudes for the unstable (inverted) potential but they don't directly affect physics at any particular real value of energy.
| {
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Why egg cooks slowly in mountains? A quick Google tells me "Because water boils at a lower temperature at high altitudes". But I am not being able to understand this answer and fill-in the gap. Like, how does an egg cook in the first place? What does it mean when we say that water boils at lower temperature? In fact, I would have thought otherwise that since water is able to boil at low temperature itself (which will be reached sooner), the egg will be able to cook sooner.
What is wrong with this reasoning?
| Since at higher altitudes, the air pressure is lower, the boiling point of water decreases, since it's easier for the energy insde the water to get free.
When A liquid starts to boil, you reach a critical point where the liquid loses a lot of heat, much more than when not boiling, thus requiring much more energy for the same increase in temperature, and lowering the equilibrium point where the flame cannot increase further the temperature of the liquid.
An egg cook because the heat transform the proteins inside, tangling them with each other. But since the boiling water has a lower temperature, this process is slightly slower.
| {
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The Energy-Momentum Tensor and the Ward Identity I have a question regarding a homework problem for my quantum field theory assignment.
For the purposes of the question, we can just assume the Lagrangian is that of a real scalar field:
$$\mathcal{L}=\frac{1}{2}\left( \partial _\mu \phi\right) ^2-\frac{1}{2}m^2\phi ^2$$.
The first part of the problem was to calculate the energy-momentum tensor corresponding to the translation symmetry $x\rightarrow x+a$. I obtained:
$$
T_\mu ^\nu =(\partial _\mu \phi )(\partial ^\nu \phi )-\delta _\mu ^\nu \mathcal{L}.
$$
So far, so good (at least I hope so).
The second part of the problem states: "Write the corresponding Ward Identity." What does he mean by this? I looked up "Ward Identity" in the index of our text (Peskin and Scroeder), but was unable to find something that seemed relevant to the problem. Where do I even begin?
| The Noether current is a conserved quantity in classical field theory. In quantum theory, it is mirrored by the Ward identity. You may possibly find this formula looking into the derivation for Schwinger-Dyson equations.
It is derived in chp 22 of Srednicki's book: http://web.physics.ucsb.edu/~mark/qft.html.
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Classical car collision I have a very confusing discussion with a friend of mine.
2 cars ($car_a$ and $car_b$) of the same mass $m$ are on a collision course. Both cars travel at $50_\frac{km}{h}$ towards each other.
They collide. Ignoring any shreds and collateral damage, what is the speed of collision that the driver of $car_a$ felt?
What I mean is, if $car_a$ were to be driven into an infinite mass wall, what would the velocity be to replicate the damage caused by the initial collision?
| I think you are asking how much damage would be done to the driver in the two cases you described. If that is your question, then the single car that is driven at a speed of $50\frac{km}{hr}$ into an infinite mass wall would experience the same damage as two identical cars being driven exactly head on at a speed relative to the ground of $50\frac{km}{hr}$ into each other.
You can easily understand this if you imagine an infinitely thin wall between the two colliding cars at exactly the plane where they collide. Assuming no shards or other pieces breaking off and going through this imaginary wall you can see that this is exactly the same as having an infinitely massive wall in place of either car since everything is exactly symmetric about this infinitely thin imaginary wall.
To be absolutely correct, this answer would actually require that the cars be left-right symmetric so that the centers of mass exactly line up perpendicular to the plane of the collision. If they were asymmetric it would be similar to symmetric cars hitting slightly off center - so there would be some torque around the point on the collision plane where the line determined by centers of mass intersect the plane. Clear?
EDIT: A situation which is equivalent to 2 cars hitting head on at 50kmph is the following: one car sitting stationary (with brakes off) while the other car hits it head on at 100kmph. This assumes a perfectly inelastic collision so that the two cars will then proceed (joined together) in the original direction of the 100kmph car but they will both be going at 50kmph. In both of these cases the change in speed of each car is 50kmph in a short time so the damages will be equivalent (either 50->0, 100->50 or 0->-50). However, if a car that is traveling at 100kmph hits an infinite mass wall the change in speed in a short time will be 100kmph so it is not equivalent to the two cases.
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What is lepton number? What exactly is a lepton number of a particle? With the charge (eg proton is just 1, not the exact charge), I can understand because it's a physical property, put a particle with charge + next to another particle with charge + and they will repel.
What is the lepton number in similar terms? Or is it just a convention that worked in the observable particle interactions (that it is conserved, like charge)
| Electric charge is a "special" kind of physical property because it corresponds to a very simple physical effect. But that's not true of most physical properties. The lepton number doesn't have any force associated with it, the way electric charge does, because it's not a coupling constant.
Lepton number is just a mathematical expression of what it means to be a lepton. The quantum fields which correspond to the particles we call leptons (electron, muon, tau, and their corresponding neutrinos) each have a lepton number of 1, and the fields corresponding to their antiparticles each have a lepton number of -1. Lepton number is considered to be a useful property because it is conserved in all observed reactions.
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Can there be black light? I mean is it possible to devise a machine that outputs darkness? I understand there are various colours that light can have. But i was wondering why there is no 'black' light. What is the logical explanation for this? I mean I am expecting an answer that goes beyond mentioning the spectrum details. All I could think of was a machine as powerful as a blackhole; it could bend the light so hard that all we would see is darkness. But is there any other way?
P.S. I am a programmer and didn't study much Physics beyond high school. This question is not a goof. I am not asking this question for fun. I seriously have this curiosity.
| It may be worth looking into an alternative theory of colour eg. Goethe. He proposes that colour arises from the interplay of light and darkness. Darkness, then, is not the absence of light but the polar opposite. It is my opinion that human consciousness is not ready to arrive at a true understanding of the phenomenon of darkness. We are still grappling with the mystery of light which remains inextricably linked with the phenomenon of fire and the activities of stars - our sun being the most obvious example. Some people believe that thought and consciousness emit a subtle light. It is the earth's atmosphere that possibly provides the most clues in the search for a better understanding of this topic.
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Quality factor for a quantum oscillator? I've been reading papers about nanomechanical oscillators, and the concept of quality factor often pops up. I understand to some extent about Q factor in classical sense, but since nanomechanic oscillators are often treated quantumly, what does Q factor mean then?
| If you have a hydrogen atom in empty space in the a p-state and ask about the transition to the ground state, Quantum Mechanics tells you that the coupling between the states is given by something called Fermi's Golden Rule. There is also something called the Einstein A coefficient which gives the probability per unit time of spontaneous emission. Since we said the atom is in empty space, that is the only transition possible, so it all somes to the same thing.
Since there is a frequency of the transition, and since there is a total elapsed time (to a power of 1/e), you can take the product of these which is basically the number of oscillations during the transition. That's the quality or Q-factor.
This all makes perfect sense in a very classical way. If you look at the formula for calculating the coupling between the states, you will see that there is something which basically comes to the dipole moment of the superposition of the s and p states. In fact, when those states are in superposition, they represent an oscillating dipole. This is well known and you can see it demonstrated in many applets such as this one by Paul Falstad The thing not everyone knows is that if you interpret this wave function as charge density, then it has electromagnetic properties that are readily calculated by Maxwell's equations. One such property is the rate at which power is being radiated classically. If we divide this rate into the total energy of the state, we get a characteristic lifetime.
It should come as a surprise to nobody that this is the same characteristic lifetime given by the Einstein A coefficient. The Einstein B coefficients, and of course the Q factors, are similarly calculable. In otherwords, the radiative properties of Schroedinger's hydrogen atom are given exactly by applying Maxwell's equations. There is no need to assume that energy is absorbed or given off in discrete lumps.
| {
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What does transport equation represent in terms of physical quantities? In my math course we're taught to solve PDE (partial derivative equations) like transport equation:
$$
c\frac{\partial u}{\partial x} +\frac{\partial u}{\partial t}~=~0.
$$
If $u(x,t)$ is the quantity transported and $c$ has speed dimension (according to my book), $\frac{\partial u}{\partial t}$ must be speed too. What does $\frac{\partial u}{\partial x}$ represent? Does anybody have a good physical example to help me understand?
| $\frac{\delta u}{\delta t}$ does not always have the dimension of speed. It is the change rate of physical quantities respect to time, $u$ can be mass or concentration of electric charge (density) or probability density $\rho$ in quantum physics.
So if we only consider the classical physics (i.e. heat conduction can be described using this function), the $\frac{\delta u}{\delta x}$ can be looked as gradient of physical quantity (in one dimension in this equation). Since there is such a gradient, therefore, we can think that this gradient will produce a "force"(not quite an actual force usually) to drive the transport. Therefore, this quantity will have a change rate respect to time. According to the conservation law, this change rate must be equal to the gradient times a constant $c$, and this $c$ has a dimension of speed.
| {
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Chiral anomaly and decay of the pion I am told that if all classical symmetries were reflected as quantum symmetries, the decay of the neutral pion $$\pi^0 ~\longrightarrow~ \gamma\gamma$$ would not happen. Why would the conservation of the axial current in QED prevent the decay of the pion? What is the non-conserved charge in this decay?
| Neutral Pion would not decay (in Your discussed case) only if the constituent quarks forming the neutral pion were massless. The professional answer to Your question can be found here:
http://www.scholarpedia.org/article/Axial_anomaly .
It clearly states (between Eq. 14 - 15 and Eq. 24 - 25), that zero mass of Fermi-Dirac field is required for the axial current conservation at classical level. If that is the case, quantum effects still allow the pseudoscalar bound state (particles) to decay into two photons due to the quantum anomaly.
However, quarks are not massless, and neutral pion can decay even at classical level into two gammas, in the same way, as para-positronium (e+e-) quantum state annihilates = decays into two gammas. Nobody speaks about anomalies in the para-positronium case, since mass of electron is clearly 0.5 MeV. In this sense, axial anomaly role in the decay of neutral pion may seem to be overestimated. R.Jackiw says (physical pion's mass can be accurately described as "approximately" vanishing) in his explanation (http://www.scholarpedia.org/article/Axial_anomaly). Neutral Pi0 mass is 135MeV, which is 135x more massive than the Positronium is. If one finds 135 MeV to be approximately vanishing, than Pi0 decay can be related to the quantum anomaly, I think.
| {
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Total Resistance of Infinite Resistor Grid? The problem of the infinite resistor grid is very common. The solution for the resistance between any 2 nodes in an infinite resistor lattice is all over the internet.
My question is somewhat similar but more pragmatic.
If we had a grid that was very large but yet finite... Then what would be the average voltage drop across a given grid for a given current density?
For arguments sake, a grid in the region of say 4000 by 4000. Maybe it would be safe to assume an infinite grid(?)
Very interesting Q. Can anyone shed any light?
| The total resistance of the grid is infinite when the grid is two dimensional and large.
If you place two point probes at location x and y on an infinite 2-d resistor grid, and impose the voltage V(x)=1 and V(y)=0, the potential obeys the discretized Laplace equation: V(up) + V(down) + V(left) + V(right) - 4 V(center) = 0 with the boundary conditions at the two given points and V=0 at infinity (beyond x and y).
In the limit that x and y are far apart, the discrete Laplace equation might as well be the continuous Laplace equation, and the solution goes like C log(|r-x|/|y-x|), so that the potential difference for any finite C diverges with the distance. This means that C has to go to zero in the large |x-y| limit, so the current vanishes. The same is true in 1d, where a line of resistors has a current which vanishes as 1/L, so the total resistance goes as the total length L. In two dimensions, the total resistance blows up as log(L).
For a three dimensional grid and higher, you do have a finite resistance for a block. Whether the limiting resistance is finite or infinite is the same problem as the recurrence/nonrecurrence of a random walk on the grid.
If you make a pseudo two-d grid using N parallel lines of N resistors in series, then the total resistance is N on each path, but there are N parallel paths, so the total resistance is R, independent of the size. This is not the same as the 2-d resistor grid, because in the 2d grid there is resistance to going vertically a long way which is similar to the resistance to going horizontally, so the horizontal resistor paths are not parallel. If you make all the vertical resistors zero, and make the separation between x and y horizontal, and make the vertical width equal to |x-y|, you recover the series/parallel situation.
The series-parallel example gives intuition about why two dimensions is critical for the transition from infinite resistance to finite resistance.
| {
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superconductor levitating in earth's magnetic field?
Possible Duplicate:
Can superconducting magnets fly (or repel the earth's core)?
I've seen superconductors levitating on magnets. But is it possible for superconductors to levitate on Earth from Earth's magnetic field?
| The lift generated by magnetic field B on a superconductor of area S is:
\begin{equation}
F = \frac{B^2S}{2\mu_0}
\end{equation}
disregarding lateral forces and assuming superconducting cylinder (or similar shape) with area S at the top and bottom and height h, we need three forces to remain in the equilibrium: magnetic pressure on top, bottom and gravity force:
\begin{equation}
F_{b} - F_{t} = F_{g}
\end{equation}
denoting density of the superconductor as ρ, Earth' gravity as g and magnetic field at the top and bottom of the object as Bt and Bb, we have
\begin{equation}
\frac{1}{2\mu_0}(B_{b}^2-B_{t}^2)=\rho gh
\end{equation}
assuming the vertical rate of change of magnetic field is nearly constant and denoting the average magnetic field as B, we have
\begin{equation}
-B\frac{dB}{dz}=\mu_{0}\rho g
\end{equation}
Compare with diamagnetic levitation (superconductor's magnetic susceptibility is -1).
Now, Earth magnetic field is between 25 to 65 μT. For the derivative I have found this survey from British Columbia with upper point on the scale being 2.161 nT/m. Assuming this to be the maximum for vertical derivative we get the required density of 1.1394e-08 kg/m3. For comparison air density at the sea level at 15C is around 1.275 kg/m3, so required density is 8 orders of magnitude smaller.
Even assuming a very high vertical derivative where B goes from its maximum 65 μT to 0 on 1 m of height results in density required of 0.00034272 kg/m3.
| {
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Can Plasma Ignite? This question on Scifi.se: Why does warp plasma burn green? mentions a fictional type of plasma called 'Warp Plasma' that is capable of igniting and when it does it sprays plasma flames/gas out of pipes as if it were oil or some other form of fuel.
Wikipedia says that the sun consists of hot plasma interwoven with magnetic fields. I couldn't find anything about plasma ignition (This is what I originally thought might be a reason why stars are so bright).
Is plasma capable of igniting?
| Ignition in what sense? Chemical energy is out of the question, because plasma by definition are mostly ions unbound of their electrons, which is needed for chemical binding.
Nuclear fusion ignition, well, that is the whole point behind nuclear fusion research. So if someone finds how to "ignite" plasma, it will revolutionize the world, the inventor will become filthy rich, or, worst case, receive a Nobel prize.
There is an independent group trying to do focus plasma fusion with something that resembles huge spark plugs. I have no idea how likely it is that they will find anything interesting soon.
| {
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What is the force between two magnetic dipoles?
*
*What is the force between two magnetic dipoles?
*If I have two current loops parallel to each other with currents $I_1$ and $I_2$ and radii $R_1$ and $R_2$ a distance $z$ from each other, what is the force between them?
*What would change if they were two solenoids instead of current loops?
*Would the same hold if it was two magnets?
*Or a magnet and a solenoid?
| The equation for the force depends on how far you are from the current source.
If you are far away, then the dipole-dipole interaction formulas can be used. In that case, all of the configurations will give the same force if they have the same dipole moments. If you are closer, then more complicated specific equations must be used for each configuration.
| {
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Hamiltonian with position-spin coupling I am solving a Hamiltonian including a term $(x\cdot S)^2$.
The Hamiltonian is like this form:
\begin{equation}
H=L\cdot S+(x\cdot S)^2
\end{equation}
where $x$ is the position operator, $L$ is angular momentum operator, and $S$ is spin operator. The eigenvalue for $L^2$ and $S^2$ are $l(l+1)$ and $s(s+1)$.
If the Hamiltonian only has the first term, it is just spin orbital coupling and it is easy to solve. The total $J=L+S$, $L^2$ and $S^2$ are quantum number. However, when we consider the second term position and spin coupling $(x\cdot S)^2$, it becomes much harder. The total $J$ is still a quantum number. We have $[(x\cdot S)^2, J]=0$. However, $[(x\cdot S)^2,L^2]≠0$, $L$ is not a quantum number anymore.
Anybody have ideas on how to solve this Hamiltonian?
| I would solve it using a matrix representation.
If we multiply Pauli matrices by $\frac{\hbar}{2}$ we can work in the following basis:
$$|n;S_z = +\rangle, |n ; S_z = -\rangle$$
Note that
$$S\cdot L = S_xL_x +S_yL_y +S_zL_z$$
$$X\cdot S = XS_x$$
$$[L,S] =0$$
You get some matrix in the $S_z$ base ($2\times 2$) and diagonalize it (finding eigenstates and eigenvalue).
| {
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What are some interesting calculus of variation problems? That I could create as a classical mechanics class project? Other than the classical examples that we see in textbooks (catenary, brachistochrone, Fermat, etc..)
| While studying classical mechanics I did the following simulation:
*
*Consider a motion in Coulomb potential: $U(r) = \frac{\alpha}{r}$
*Fix starting and final points $p_1$ and $p_2$, and consider different paths in a form: $$p_1 + (p_2 - p_1)\lambda + \vec{a}\sin(\pi\lambda) + \vec{b}\sin(2\pi\lambda) + \vec{c}\sin(3\pi\lambda)$$
Where $\lambda$ is the parameter along our path and $\vec{a},\vec{b},\vec{c}$ are 2D vectors, that parametrize it.
*Take some initial parameters $(\vec{a},\vec{b},\vec{c})$ and calculate the action along the path by means of Maupertuis' principle.
*Make a small random change in $\vec{a}' = \vec{a} + \mbox{random },\vec{b}' = \vec{b} + \mbox{random }$ and $\vec{c}' = \vec{c} + \mbox{random}$.
*Calculate the action for $(\vec{a}',\vec{b}',\vec{c}')$ parameters. If action becomes smaller -- replace the parameters with new values $\vec{a}=\vec{a}',\vec{b}=\vec{b}',\vec{c}=\vec{c}'$.
*Goto step 4.
Here is what I've got in the end:
Here $\alpha = -200, p_1 = (0,-5)$ and $p_2 = (0.17,-0.17)$.
Numbers on top are: left ("Шаг") -- is a step number in the simulation, right ("Действие") -- is the value of the Maupertuis' action.
Red and green lines are real trajectories in the potential and the black line is my "test trajectory". So one can see that the simple random walk in parameter space can find some of the real paths of the body.
| {
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Is it possible to recover Classical Mechanics from Schrödinger's equation? Let me explain in details. Let $\Psi=\Psi(x,t)$ be the wave function of a particle moving in a unidimensional space. Is there a way of writing $\Psi(x,t)$ so that $|\Psi(x,t)|^2$ represents the probability density of finding a particle in classical mechanics (using a Dirac delta function, perhaps)?
| The short answer: No, does not exist any wavefunction in Hilbert space which reproduces classical mechanics.
The classical limit of quantum mechanics is studied with some deep in Ballentine textbook. For instance, section 14.1 is devoted to the Ehrenfest theorem and it is shown that the theorem is neither necessary nor sufficient to define the classical
regime.
The paper What is the limit $\hbar \rightarrow 0$ of quantum theory? (Accepted for publication in the American Journal of Physics) shows that Schrödinger's equation for a single particle moving in an external potential does not lead to Newton's equation of motion for the particle in the general case. Page 9 of this more recent article precisely deals with the question of why no wavefunction in the Hilbert space can give a classical delta function probability.
| {
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Physical laws prior the big bang (quantum fluctuations) A theory among scientists says that quantum fluctuations caused the big bang and created the universe. This seems plausible to me.
What I can't grasp yet is how a quantum fluctuation can even start without an existing universe. Aren't all physical laws created with the universe? I understand that there is no notion of "before" with respect to time, however the big bang is theorised to have occurred, but for that to occur there must have first existed something right?
I wonder also, if there was a more nothingness instead of vacuum before the universe existed and how a quantum fluctuation could have started really from ex nihilo instead of a vacuum.
| Indeed, there exist some pre-big-bang theories that arose from string theory. A notable name in this area is Gabriele Veneziano. You can find some information here. This article is somehow technical but should convey the right flavor of these ideas. As you can read from this, there is a two-dimensional space where strings live and a target space, the one we live on, that should start, with inflation. Quantum fluctuations that start the big-bang are happening on the two-dimensional space. These theories have some problems, e.g. the graceful exit problem from the inflationary phase, but are an active field of study.
| {
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This Expansion-of-the-Universe-Diagram Confuses Me The following blue-cone Wikipedia diagram confuses me.
At any point of cosmological time the encircling horizontal lines in the diagram are of finite circumference. That is indicative of a closed model of the universe.
Queries:
1. Why does the author use a closed model of the universe to explain his point?
2.Can we conclude "It is also possible for a distance to exceed the speed of light times the age of the universe, which means that light from one part of space generated near the beginning of the Universe........." if we draw the same diagram on a flat sheet of paper[instead of using the cone we take a flat surface],remembering that the null geodesics are always straight lines in the flat spacetime context?
| If you look at the top view of the cone in the Wikipedia article you linked to, you'll notice that it's not actually a closed cone. So this diagram represents the evolution of a 1D open curve in a timelike slice of spacetime, not the whole universe. It doesn't say anything about instantaneous large-scale structure.
Even if the cone were closed, you could still look at it as representing the evolution of a 1D closed curve in spacetime, which does not necessarily have to wrap around the universe. That way, the diagram could be accurate even if the universe is open.
| {
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Is it necessary to embed a 4D surface in 5D space? Lets consider the line element:
$$ds^2=dr^2+r^2[d\theta^2+\sin^2\theta d\phi^2]$$
There are three variables r,theta and phi.
If we use a surface constraint like r=constant the number of
independent variables is reduced by one--now we have two independent
variables.These surfaces[corresponding to r=const] may be embedded in
a three dimensional space.
Now lets consider Schwarzschild's metric:
$$ds^2=(1-2m/r)dt^2 - (1-2m/r)^{-1} dr^2 - r^2[d \theta^2+\sin^2 \theta d\phi^2]$$
If we use a surface constraint[for example: t=constant] we have three
independent variables.The resulting time slices are three dimensional surfaces which are naturally embedded in a 4D space.
The General Relativity metric has four variables: one relating to
time and three relating to the spatial coordinates.Any surface
constraint would reduce the number of variables to three.
In fact any arbitrary spacetime curve[world-line] may be made to lie
on a 3D- Surface obtained by applying some suitable constraint on 4D
space.The constraint may not be a simple one like t= constant or r= constant. It may be of a complicated nature.
For the purpose of embedding in GR a 4D space seems to be sufficient.
Queries:
*
*Is it essential the we should consider a 4D surface embedded in 5D space to understand or interpret GR?
*It appears that the curve[4D path] is more important than the surface on which it is lying since the same curve may be made to lie on several distinct surfaces at the same time[you may extend this to the case of 4D surf embedded in 5D space].Is this interpretation correct?
| Any enclosing space is outside of the problem domain of GR: All results can be obtained from within the space-time. Physically, it makes no sense to talk about an enclosing space which has no impact whatsoever on measurements. In particular, even though we say space-time is curved, the question Where does it curve to? makes no sense in the framework of GR.
Also, 5 dimensions are not enough to contain arbitrary 4-dimensional pseudo-Riemannian manifolds of index 1 if you want to preserve the metric. Quoting C. J. S. Clarke, On the Global Isometric Embedding of Pseudo-Riemannian Manifolds:
The space-time of general relativity can be embedded isometrically in $E^{2,q+2}$ (pseudo-Euclidean space of signature $q-2$) where $q=46$ or $q=87$ for compact or non-compact space-time, respectively.
However, the result is only valid for finite $k$ and not $C^\infty$.
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Why do 3d spheres and gravity tend to rotating discs on one plane? Whether is it our solar system or a whole galaxy, there is usually a massive object (star or black hole) at the centre with gas and objects rotating around it.
The gravitational effect of the star/black hole extends uniformly (more or less) in every direction in 3d. Why does matter tend towards a single plane?
Furthermore, what happens to matter that approaches after the "disc" is formed when it is pulled in from anywhere off the plane, why does it join the plane rather than forming another plane?
I suspect angular momentum has something to do with it, but would appreciate a "pop science" explanation.
Many thanks
Andrew
| The way I look at this is to wonder what would happen if things weren't in the same plane.
*
*The obvious issue is collisions - space may be quite empty but if you have a large amount of dust orbiting in different planes there will be collisions
*Gravitational forces - more likely than collisions, when bodies come near they affect each other. In a plane, the forces will also be in the plane; where two bodies are in different planes the forces will alter the orbits. This leads to two simple steady states - objects ending up in the same plane, or one being flung out of the system
*Angular momentum - even if the initial aggregate movement of all the particles in a dust cloud is small, by the time they have fallen towards each other they would have built up spin, as angular momentum needs to be conserved. Having particles orbiting in different planes or directions will cancel out momentum, so the simplest outcome is for spin to happen in the same direction
*Drag - energy losses occur, but these are least if all objects are orbiting together
| {
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How can I prevent being zapped by static electricity every time I touch a doorknob or handle in the office? I don't know what it is about this office, but it seems everything I touch (doorknob, bathroom faucet, edge of kitchen sink in the break room), I get zapped by static electricity. It's getting old. I feel like that scene in Office Space.
I've worked in other offices and it's not nearly this crazy. This has been going on for months so it's not the weather.
Why does this happen and is there any way I can defend against this evil?
| Touch it with a key or something metal first.
| {
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How should one combine the uncertainties from the ATLAS and CMS measurements? First off, a naive theorist question - How are measurements divided between the different detectors at the LHC? I would imagine that for a short run time, say, the CMS detector is active and all the others are turned off (the beam just passes through) and the beam is directed so as to collide in the CMS detector. Then for another run period, ATLAS is activated and collisions happen only there, giving another set of data. This would mean that the measurements at ATLAS and CMS are completely independent. Is this correct?
Given that, maybe I missed it, but I never saw anyone quote a combined average and uncertainty for both the ATLAS and CMS excesses. If, let's say, there's a 1 in 300 chance that the CMS excess is a statistical fluke and 1 in 200 that the ATLAS excess is a statistical fluke, as someone with a very limited understanding of probability and statistics, I would just multiply the probabilities to see how likely it isn't a Higgs signal and looking at the small value, would be very excited. How does one go about rigorously calculating the combined likelihood that the excess is not a Higgs signal?
| Have a look at the recent discussions where the experiments are combined:
http://motls.blogspot.com/2012/02/higgs-signal-grew-from-38-to-43-sigma.html
and error estimates discussed :
http://blog.vixra.org/2012/02/10/some-technical-points-about-combining-sigmas/
Here are the recent individual ones:
http://www.science20.com/quantum_diaries_survivor/atlas_and_cms_publish_2011_higgs_results-86735
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Can you magnetize iron with a hammer? We know that a piece of ferromagnet, such as iron, can be magnetized by putting in a strong magnetic field to get domains parallel to the field grow.
I also remember from pop. culture and MacGyver old tv series that you can magnetize a piece of iron by hitting it hard, with a hammer say, along the same direction.
1-Is this way of magnetizing iron scientific? or is it pseudoscience?
and if it is scientific then:
2-what is the physical principle that will allow iron to get magnetized by hitting? and
3-how about nonferromagnetic materials?
| Seems that it can be done, and here are instructions
Copying from the link
Strike an iron nail squarely and sharply several times with a hammer while keeping the nail positioned in a north-south orientation. The impact of the hammer with the iron nail causes the magnetic domains within the nail to break loose from their current orientation. The Earth's magnetic field will then reposition the domains into a new orientation parallel with the Earth's magnetic field.
It is evident that this can be done only with materials that have small domains with magnetization, which are randomly oriented, so the material has to be ferromagnetic.
| {
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How does gravity work underground? Would the effect of gravity on me change if I were to dig a very deep hole and stand in it? If so, how would it change? Am I more likely to be pulled downwards, or pulled towards the edges of the hole? If there would be no change, why not?
| Assuming spherically symmetric mass distribution within Earth, one can compute gravitational field inside the planet using Gauss' law for gravity. One consequence of the law is that while computing the gravitational field at a distance r < R (with R being the radius of the Earth), one can ignore all the mass outside the radius r from the center
\begin{equation}
\oint_{S_r} g_r \cdot dA = -G \int_{B_r} \rho dV
\end{equation}
where gr is the gravitational field at distance r from Earth's center, ρ is Earth's density, Sr is the sphere of radius r centered on Earth's center of mass and Br is the volume enclosed by Sr. Assuming that ρ only depends on the distance r from the center of the Earth, we can simplify this as follows
\begin{equation}
\oint_{S_r} g_r \cdot dA = -4\pi G \int_0^r \rho(s) ~s^2ds
\end{equation}
\begin{equation}
g_r = -\frac{G}{r^2} \int_0^r \rho(s)~s^2ds
\end{equation}
Setting Mr to denote the portion of Earth's mass enclosed within Sr, we can rewrite the last formula as
\begin{equation}
g_r = -\frac{GM_r}{r^2}
\end{equation}
Now, letting ρr denote the average density of the portion of the Earth enclosed within Sr, we have
\begin{equation}
g_r = -\frac{4 \pi G \rho_r r}{3}
\end{equation}
The conclusion is that the gravity inside Earth depends roughly linearly on the distance from the center of the planet and density variations account for the deviations from linearity.
An interesting way to visualize this is to think of an over 12,700 kms long elevator from Hamilton, New Zealand to Cordoba, Spain. During the travel (which at average speed of 200km/h would take almost three days) passengers would feel gradual and roughly linear decrease in weight and in the middle of the journey would experience weightlessness followed by gradual increase in weight as they near the surface on the other side. Also, around the midpoint of the journey the floor and the ceiling would swap.
| {
"language": "en",
"url": "https://physics.stackexchange.com/questions/18446",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "56",
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Are proton, antiproton, electron, positron the only observed subatomic particles that can freely exist and don't decay, i.e. are stable? Are proton, antiproton, electron, positron the only subatomic particles that can freely exist (i.e. I don't want particles that only exist in bound state as constituents such as quarks) and don't decay, i.e. are stable?
What about muons?
Are there other particles/hadrons that can exist freely (i.e. not in some bound states) and don't decay?
| No. At a minimum neutrinos and anti-neutrinos are also stable{*} and exist in unbound systems.
Additionally the electromagnetic carrier boson (i.e. the photon) can exist for arbitrarily long times in the reference frames of massive objects (it's proper time is necessarily zero). The same could be said for the graviton if we had experimental confirmation of it's existence.
Further, many beyond-the-standard-model candidate theories feature (anti-)proton decay, though the current experimental limits require this to be a very slow process indeed.
{*} We have to be a little careful about what we mean here. The pure mass states $\nu_i$ are stable in free propagation, however neutrinos are created and destroyed in flavor states. So we have time-dependent superpositions of mass-states, in which the sum of the lepton flavor numbers is conserved.
| {
"language": "en",
"url": "https://physics.stackexchange.com/questions/18498",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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What does a unitary transformation mean in the context of an evolution equation? Let be the unitary evolution operator of a quantum system be $U(t)=\exp(itH)$ for $t >0$.
Then what is the meaning of the equation
$$\det\bigl(I-U(t)e^{itE}\bigr)=0$$
where $E$ is a real variable?
| If the dimension of the state space is finite, say $n$, then your question makes sense since the determinant makes sense.
Now suppose that $E$ is a real number such that (for all $t$) $$\det (I-U(t)e^{itE} ) =0.$$ Your equation implies that $I-U(t)e^{itE}$ is not invertible (if it were invertible, its determinant would be non-zero). This implies that there exists a non-zero vector $v$, a state, for which
$$v=U(t)e^{itE}v$$ and hence $e^{-itH}v=e^{itE}v$ and hence $$ Hv = -Ev$$ so $v$ is an eigenvector with eignevalue $-E$, i.e., $-E$ is an energy level of the system and when it is in the state $v$ it will, if measured, produce the result of energy$ = -E$ with probability one.
But in infinite dimensions you can't be so direct.
| {
"language": "en",
"url": "https://physics.stackexchange.com/questions/18539",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Effects of a Coiled Cable Okay, I've got a little bit of a layman's question here.
We're doing a bit of spring cleaning in our office and we've found a cabinet with boxes upon boxes of stored wires, so naturally, this discussion arose...
Picture a normal, bog-standard wire, with a plastic outer coating. Now, quite often when these wires are stored, they will wrapped up and twisted, to effectively make a coil.
I was just wondering what the effects of this type of storage would have.
What if you had a 15m wire and only used the each end to cover about a single meter (leaving 14m still twisted and wrapped in the middle), what the effect of the electrical current running through this have?
Thanks for helping us settle a mild dispute!
| You are talking about the inductive effects of the coil of wire. Essentially a wrapped up coil of metal with electrons running through it creates a linear magnetic field since moving electrons through a wire creates a redial field and if you approximate the coil to have infinite loops the field becomes liner.
But, this effect would be very, very small for the wires you are talking about since (a) the coils are not very densely packed and (b) not very much current is flowing through them
Here is the wiki on inductors:
http://en.wikipedia.org/wiki/Inductor
The simple relationship between voltage($v$), inductance($L$), and current($i$) is:
$$V(t) = L \frac{di(t)}{dt}$$
One last thing to consider, magnetic field drops off with distance fast (so as you move away from the source it gets really weak). The plastic protective covering around your wires are a relatively similar thickness to the wires themselves and would buffer anything nearby to most the magnetic effects (which would be weak to begin with)
| {
"language": "en",
"url": "https://physics.stackexchange.com/questions/18623",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
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Spin angular momentum of a system of particles : Is there any energy associated with it? Consider a system of point particles , where the mass of particle $i$ is $μ_i$ and its position vector is $\vec{r}_i$. Let $\vec{r}_\text{cm}$ is the position vector of the center of mass of the system. Considering the system from a reference frame attached to the center of mass, the system may have a spin about the center of mass and it is given by the spin angular momentum $\vec{L}_{spin}$. It is given by the expression
$$\vec{L}_{spin} = \sum_i \mu_i \Bigl[(\vec{r}_i - \vec{r}_\text{cm}) \times (\dot{\vec{r}}_i - \dot{\vec{r}}_\text{cm}) \Bigr]$$
The rate of change of this spin angular momentum is the total torque acting on the system about the center of mass in the center of mass reference frame.
My question is, is there any (spin kinetic (may be)) energy associated with the spin angular momentum in the center of mass reference frame ? How is it defined ?
| The energy associated with the internal rotations is the rotational energy
http://en.wikipedia.org/wiki/Rotational_energy
given by the formula
$$ K = \frac 12 I \omega^2 $$
where $\omega$ is the angular frequency and $I$ is the moment of inertia with respect to the axis of rotation
http://en.wikipedia.org/wiki/Moment_of_inertia
$$ I = \int {\mathrm d}m \, \rho^2 $$
where $\rho$ is the distance of the infinitesimal mass ${\mathrm d}m$ from the axis of rotation. Also,
$$ I = I_{ab} n_a n_b $$
in terms of the tensorial moment of inertia $I_{ab}$; $\vec n$ is the unit vector along the axis of rotation.
| {
"language": "en",
"url": "https://physics.stackexchange.com/questions/18715",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Elasticity of Space; How does the expansion of Space affect gravity? Does space have an elastic quality?
What I was thinking about was if space is expanding, is it being 'stretched', like a balloon being blown up, and if so, is this causing gravity to weaken? Imagine space as a 2 dimensional sheet (got this from one of Brian Greene's books) with planetary bodies resting on it and causing a depression in it, if you were pulling this sheet from all sides over a period of time, you would cause the depression of the planetary body to decrease and eventually become flat, which if we go back to reality, would mean that the gravitational 'constant' had changed to the point where the planetary body had no influence on those objects which were previously orbiting around it (or even residing on it's surface).
Is this the case in reality? Or does space not have an elastic quality? If not, can you explain to me what exactly it means for space to be expanding?
In case you didn't notice, I'm a layman (hence the Brian Greene books :p), so try to keep your answers/explanations conceptual if possible.
| But, how do gravitational waves propagate unless there were some form of fluidity to the space between bodies? There is a very fluidic like behavior of wht we refer to as the "fabric" of spacetime. Even with local large mass gravitational distortion, once the object moves from a given reference point the fabric returns to its previous contour like a boat through water. It may not be fluid by classical definitions, but...
| {
"language": "en",
"url": "https://physics.stackexchange.com/questions/18768",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 4,
"answer_id": 2
} |
Simple applications of group theory which can be understood by a senior undergrad I am looking for references (books or web links) which have "simple" examples on the use of group theory in physics or science in general.
I have looked at many books on the subject unfortunately they usually require extensive technical coverage of the basics, i.e. the 1st 100 pages or something, to be able to start discussing applications.
I believe that there is an easy way to explain anything (it's just hard to find it).
| I recommend " The Theory of Groups and Quantum Mechanics " by Hermann Weyl http://books.google.co.uk/books?id=jQbEcDDqGb8C&printsec=frontcover&source=gbs_ge_summary_r&cad=0#v=onepage&q&f=false
Although the book is written in a old school way, I found it interesting!
| {
"language": "en",
"url": "https://physics.stackexchange.com/questions/18909",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 3,
"answer_id": 1
} |
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