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<p>As we know the predominant theory where does the moon come from is that a Mars size planet hit the earth and took a chunk out of it which eventually materialized into moon.</p>
<p>My question is that if a Mars size object were to hit Earth, wouldn't it knock it off the orbit all together? What kind of collision is required to knock a planet of its orbit. By 'knock off' I mean it would alter the orbit of Earth and possibly speed so that it will not have stable orbit anymore so it will either (gradually) leave solar system or (gradually) collapse into sun.</p> | 4,463 | [
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<p>I've been reading about black holes, and I keep coming across the fact that time runs slower near a them. My questions are: Does this mean that if you left Earth at age 30 and spent 30 years near the black hole, then you would be 45 years old? Do you get older more slowly, because if it's all the same what is the point of travelling near a black hole?</p> | 143 | [
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<p>Imagine a photo that is taken of someone looking straight at the camera.</p>
<p>Why when we look at the photo now <em>from any direction</em> it looks as if the person is looking straight at us?</p> | 4,464 | [
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<p>I apologize in advance for possible errors in my premises as I have no precise knowledge of Maxwell equations. Proposals for the correction or even abandon of my question are welcome.</p>
<p>As Maxwell equations are a full description of electromagnetic waves I suppose that they also describe the <strong>time</strong> a wave takes from the place of its emission to the place of absorption. My question: As Maxwell equations are relativistically invariant (or at least compatible with special relativity), do they also yield the <strong>proper time</strong> (which is always zero for electromagnetic waves in vacuum)?</p> | 4,465 | [
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<p>Our house has a glass sliding door to the shower. The shower has the dimensions of about 2 feet wide, 5 feet long, and 6 feet high. Above the door (and shower head) there is about 1 foot of open space before the ceiling to let the heat escape.</p>
<p>Sometimes I forget to slide the door all the way to the end of the other side, leaving about a 1 inch space in between the door and the wall. Cold air comes through so much that I <em>always</em> known when there is a crack, the amount of cold air I feel does not seem proportional to the opening at all. I would think that the amount of heat being produced by the shower would be enough to where I don't feel the cold air, but that does not seem to be the case.</p>
<p>What I'm thinking is that it is not that the heat is leaving through there, heat is escaping through the top since heat rises, but rather the cold air is coming through the opening due to a difference in pressure. Is this the case? How come it does not seem proportional to the opening?</p>
<p>How come I can feel a fair amount of cold air coming from the small opening at the opposite end of the shower? </p> | 4,466 | [
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<p>I'm trying to find the Green's functions for time-dependent inhomogeneous Klein-Gordon equation which is :
\begin{align*}
\left[ - \nabla ^2 + \frac{1}{c^2} \dfrac{\partial ^2}{\partial t^2} + \kappa ^2 \right] \psi({\mathbf{r},t )} = \rho(\mathbf{r},t)
\end{align*}</p>
<p>It has been mentioned in the question that I can find the Green's functions :
\begin{align*}
&G_R(\mathbf{r} , t , \mathbf{r'} , t') = \dfrac{c}{8 \pi ^2 \mathbf{R} i }\dfrac{d}{d\mathbf{R}} \int_{- \infty}^{+\infty} \dfrac{e^{ i \frac{R}{c} \sqrt{q^2 - k^2 c^2}}}{\sqrt{q^2 - k^2 c^2}} e^{ - iq (t - t')} dq
\\
&G_A(\mathbf{r} , t , \mathbf{r'} , t') = -\dfrac{c}{8 \pi ^2 \mathbf{R} i }\dfrac{d}{d\mathbf{R}} \int_{- \infty}^{+\infty} \dfrac{e^{- i \frac{R}{c} \sqrt{q^2 - k^2 c^2}}}{\sqrt{q^2 - k^2 c^2}} e^{ - iq (t - t')} dq
\end{align*}
using the fourier transform, but when I use the fourier transform I don't gain the proper answer.
The fourier transform which I use is the one which is generally given as :
\begin{align*}
f(r) = \dfrac{1}{\sqrt{2 \pi}} \int_{- \infty}^{\infty} e^{ik.r}\hat{f}(k) dk
\end{align*}
but from this transform I cannot find $G_A$ and $G_R$.</p>
<p>Is there another transform which I should use to find the Green's functions? </p>
<p><strong>Edit</strong>
The Green's function which I wind up with is :
\begin{align*}
G_A(\mathbf{r} , t , \mathbf{r'} , t') = \dfrac{1}{(2\pi)^4} \int d^3\mathbf{k} dk' \frac{1}{k^2} e^{i\mathbf{k}.(\mathbf{r} - \mathbf{r'})}e^{ik'(t-t')}
\end{align*}
which is not even similar to the answer given here!</p> | 4,467 | [
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<p>when the sound barrier is broken, a series of concentric waves of sound is produced.Does it mean when the speed of light barrier is broken, a ripple of photons are created in the space-time fabric?</p> | 4,468 | [
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<p>I am reading Itzykson and Zuber's Quantum Field Theory book, and am unable to understand a step that is made on page 246:</p>
<p>Here, they consider the elastic scattering of particle $A$ off particle $B$:</p>
<p>$$A(q_1) + B(p_1) ~\rightarrow~ A(q_2) + B(p_2)$$</p>
<p>and proceed to write down the $S$-matrix element using the LSZ formula, with the $A$ particles reduced:</p>
<p>$$S_{fi}=-\int d^4x\, d^4y e^{i(q_2.y-q_1.x)}(\square_y+m_a^2)(\square_x+m_a^2)\langle p_2|T \varphi^\dagger(y) \varphi(x)|p_1 \rangle \tag{5-169}$$</p>
<p>Then they say that because $q_1$ and $q_2$ are in the forward light cone, the time-ordered product can be replaced by a retarded commutator:</p>
<p>$$T \varphi^\dagger(y) \varphi(x) ~\rightarrow~ \theta(y^0-x^0)[\varphi^\dagger(y),\,\varphi(x)]\,.$$</p>
<p>This justification for this replacement completely eludes me. What is the mathematical reason for this?</p> | 4,469 | [
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<p>Suppose a spring with spring constant 1 N m^-1 is held horizontally. If a pull of 1 N is applied to its left end and a pull of 2 N is applied to its right end, how much longer would the spring be as compared to its unstretched length?</p>
<p>I came up with this question because the spring related qurstions I've seen all have one end fixed some where. I am totally stuck on this one and I really have no idea where to start. </p>
<p>All I can figure out is that the spring will accelerate due to a net force. It seems inappropriate to be taking net forces, since pulling both ends with a larger force makes the spring more elongated.</p>
<p>P.S. this is not homework</p> | 4,470 | [
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<p>One can no nothing about the magnetic force and yet arrive at it by taking the relativistic effects of a current and a moving charge system into account. I ask whether there exists such an inherent force in case of gravity.</p> | 4,471 | [
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<p>Are there any examples of common substances whose decay is not exponential?</p>
<p>We're used to thinking about radioactivity in terms of half-lives. This is a concept that makes sense only for a decay that is exponential. However, there are plenty of physics articles on the subject of non exponential decay. It seems to be theoretically ubiquitous. For example:</p>
<blockquote>
<p>The decay of unstable quantum states
is an ubiquitous process in virtually
all fields of physics and energy
ranges, from particle and nuclear
physics to condensed matter, or atomic
and molecular science. <strong><em>The exponential
decay, by far the most common type, is
surrounded by deviations at short and
long times$^{1,2}$.</em></strong> The short-time
deviations have been much discussed,
in particular in connection with the
Zeno effect$^{3,4,5}$ and the
anti-Zeno effect$^{6,7,8,9}$.
Experimental observations of short$^{10,11}$
and long$^{12}$ time deviations
are very recent. A difficulty for the
experimental verification of long-time
deviations has been the weakness of
the decaying signal$^{13}$, but also the
measurement itself may be responsible,
because of the suppression of the
initial state reconstruction$^{2,14}$.</p>
<p>1) L. A. Khalfin, Zurn. Eksp. Teor.
Fiz. 33, 1371 (1957), English
translation: Sov. Phys. JETP 6 1053
(1958).<br>
2) L. Fonda and G. C.
Ghirardi, Il Nuovo Cimento 7A, 180
(1972).</p>
</blockquote>
<p>10.1103/PhysRevA.74.062102, F. Delgado, J. G. Muga, G. Garcia-Calderon<br>
<a href="http://arxiv.org/abs/quant-ph/0604005v2"><em>Suppression of Zeno effect for distant detectors</em></a></p>
<p>So are there any examples of deviations from long time decay? If not, then why not? Is the theory wrong or simply impractical? And is there a simple, intuitive explanation for why long decays should not be exponential?</p> | 4,472 | [
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<p>If I understand correctly, when an earthquake occurs, energy will be transferred to the water, resulting in water waves. As the waves reach seashore, because the sea depth is getting shallower and wavelength is getting shorter, the height of the wave gets push up, resulting in tsunami. In other words in deep sea, water won't get pushed up as high as the water in shallow seashore.</p>
<p>Is my understanding correct? Is there a quantitative way to express the physics behind all this?</p> | 4,473 | [
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<p>I know that there's no evidence at this point for "white holes" however would it even be mathematically possible for a black hole to be connected to a white hole (total opposite so everything would be expelled, after some really extreme physical conditions)? Maybe with a wormhole connecting them? If this was even possible (if the black hole or connection could actually be created and be stable enough), would that matter be expelled into a different universe, etc? Maybe even a different region of spacetime? Just curious, as it would be a cool idea. </p> | 4,474 | [
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<p>I've come upon Dr. J. Marvin Herndon's theory that the earth's magnetic field is generated by a hot nuclear reactor operating in the center of the earth. This is backed by various papers, some of them peer reviewed:</p>
<p><a href="http://www.nuclearplanet.com/Herndon%27s%20Nuclear%20Georeactor.html" rel="nofollow">http://www.nuclearplanet.com/Herndon%27s%20Nuclear%20Georeactor.html</a></p>
<p>His theory purports to explain various anomalies such as the <a href="http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6V66-487F9KP-3&_user=10&_coverDate=11/30/1987&_rdoc=1&_fmt=high&_orig=gateway&_origin=gateway&_sort=d&_docanchor=&view=c&_searchStrId=1680424709&_rerunOrigin=google&_acct=C000050221&_version=1&_urlVersion=0&_userid=10&md5=9a1068470e01f9f819550010d40e9418&searchtype=a" rel="nofollow">high level of He3 in basalt in Hawaii and their dependency with time</a>. A recent Herndon arXiv paper is:</p>
<p><em>Uniqueness of Herndon's Georeactor: Energy Source and Production Mechanism for Earth's Magnetic Field</em>
<a href="http://arxiv.com/abs/0901.4509" rel="nofollow">http://arxiv.com/abs/0901.4509</a></p>
<p>His papers suggest that the <a href="http://en.wikipedia.org/wiki/Natural_nuclear_fission_reactor" rel="nofollow">natural reactors sometimes seen in uranium deposits</a> must also occur at the center of the earth. He says that the conventional explanation for the earth's magnetic field fails because the <a href="http://en.wikipedia.org/wiki/Rayleigh_number" rel="nofollow">Rayleigh number</a> for the core is inappropriate as far as determining whether convection exists.</p>
<p>Since this is not accepted geology, what are the problems with the theory?</p> | 4,475 | [
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<p>How the field and interactions are changed when we assume that proton has finite radius in atom for example? What gives the finite size effect? Is it the higher moments of multipole expansion?</p> | 4,476 | [
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<p>I have to derive <a href="http://en.wikipedia.org/wiki/Wien%27s_displacement_law" rel="nofollow">Wien's displacement law</a> by using <a href="http://en.wikipedia.org/wiki/Planck%27s_law" rel="nofollow">Planck's law</a>. I've tried but I come to a unsolvable equation (well I can't solve it) anywhere I look online it comes to the same conclusion, you need to solve an equation involving transcendentals which I have no idea what they are nor any of the math classes required for this physics course teach it. Once that equation is solved, the next steps are quite simple, by using the solution.</p>
<p>Maybe my professor just wants us to google the solution to the equation and used it to keep solving the problem?</p>
<p><strong>anyway, what really bothers me is this:</strong></p>
<p>My book shows Planck's law in terms of frequency</p>
<p>$$B_\nu (T) = \frac{2h\nu^3}{c^2}\frac{1}{e^{h\nu/(k_B T)} - 1}$$</p>
<p>but to do the problem I need it in terms of wavelength</p>
<p>$$B_\lambda (T) = \frac{2hc^2}{\lambda^5}\frac{1}{e^{hc/(\lambda k_B T)} - 1}$$</p>
<p>And $v = c/\lambda$, so by that then it can be said that the exponent of $\lambda$ would be 3 when it is expressed in terms of lambda (plus $c$ would have an exponent of 1). But Wikipedia says it is 5 and not 3. It multiplies by the derivative of $\nu$, but I don't get why. I'm just substituting, not differentiating, no need to use the chain rule.</p> | 144 | [
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<p>This question is about terminology for <a href="http://en.wikipedia.org/wiki/Physical_quantity" rel="nofollow">physical quantities</a>. </p>
<p>When we talk about magnitude (while talking about scalars and vectors) do we refer to just number or Number along with units?</p>
<p>example: If a person weighs 120 pounds, then "120" is the numerical value and "pound" is the unit. </p>
<p>Which is magnitude? 120? or 120 pounds?</p>
<p><strong>EDIT:</strong> </p>
<p>In the book I'm using its written as</p>
<blockquote>
<p>The number indicates the magnitude of the scalar quantity and is inversely proportional to the unit chosen.</p>
</blockquote>
<p>This statement is wrong. Right? Its not the number alone. Its along with units. </p> | 4,477 | [
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<p>What are some "real" world applications of the particle in a box (PIB) and the finite square well (FSW) which are discussed in an intro quantum mechanics class? For instance, I know that the PIB can applied to quantum dots and the FSW to the Ramsauer-Townsend effect. How about other applications?</p> | 4,478 | [
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<p>Why does the Earth revolve with the Sun at one of its foci? Does the other focus do nothing? Why is there this asymmetry in our solar system? </p> | 4,479 | [
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<p>In the context of group theory (in my case, applications to physics), I frequently come across the phrase "the ${\bf N}$ of a group", for example "a ${\bf 24}$ of $SU(5)$" or "the ${\bf 1}$ of $SU(5)$" (the integer is usually typeset in bold).</p>
<p>My knowledge of group theory is pretty limited. I know the basics, like what properties constitute a group, and I'm familiar with simple cases that occur in physics (e.g. rotation groups $SO(2)$, $SO(3)$, the Lorentz group, $SU(2)$ with the Pauli matrices as a representation), but not much more. I've got a couple of related questions:</p>
<ul>
<li>What is meant by "${\bf N}$ of a group"?</li>
<li>Is is just shorthand for an ${\bf N}$ representation? If so, what exactly is an ${\bf N}$ representation of a given group? :-)</li>
<li>How can I work out / write down such a representation concretely, like the Pauli matrices for $SU(2)$? I'd be grateful for a simple example.</li>
<li>What does it mean when something "transforms like the ${\bf N}$"?</li>
</ul> | 345 | [
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<p>I just learned that strong and weak nuclear forces relate to decay/emission.
I know absorbtion depends on Energy levels(QM) and heat(thermodynamics , kinetic energy , entropy) and E = gamma mc^2 ( special relativity).
Assuming collision , are there fundamental forces associated with absorbtion ?</p>
<p>I add that I am not an expert in QM.</p>
<p>Edit : I noticed my question is not so good , but that was because of my ignorance. My apologies.</p> | 4,480 | [
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<p>Consider a charge conjugation operator which acts on the Dirac field($\psi$) as<br>
$$\psi_{C} \equiv \mathcal{C}\psi\mathcal{C}^{-1} = C\gamma_{0}^{T}\psi^{*}$$
Just as we can operate the parity operator on the Lagrangian, and we say that a theory has a symmetry if $$\mathcal{P}\mathcal{L}(t,x^{i})\mathcal{P}^{-1} = \mathcal{L}(t,-x^{i})$$</p>
<p>Suppose we operate $\mathcal{C}$ on the Dirac Lagrangian what should we get?
$$\mathcal{C}\mathcal{L}_{Dirac}(x^{\mu})\mathcal{C}^{-1} = \mathcal{L}^{*}_{Dirac}(x^{\mu}) ?$$ in analogy to the transformation of the scalar field $\phi$ under charge conjugation. </p>
<p>On a same note one can ask what equation should $\psi_{C}$ satisfy? Should it satisfy the conjugated Dirac equation as $$(i\gamma^{\mu}\partial_{\mu} + m)\psi_{C} = 0 ?$$ If so can someone give me the physical interpretation for it.</p>
<p>I am asking this question as I want to explicitly use $\psi_{C}$ and check whether it keeps the Dirac Lagrangian invariant. I have done a calculation by substituting $\psi_{C}$ in the Dirac equation and have found it is not satisfying as shown below. </p>
<p>$$(i\gamma_{\mu}\partial_{\mu} - m)\psi_{C} = i(\gamma_{\mu}C\gamma_{0}^{T})\partial^{\mu}\psi^{*} - (C\gamma_{0}^{T})m\psi^{*}$$
We will use $C^{-1}\gamma_{\mu}C = - \gamma_{\mu}^{T}$ and $\{\gamma_{\mu}^{T},\gamma_{\nu}^{T}\} = 2g_{\mu\nu}$.<br>
Consider
\begin{align}
\gamma_{\mu}C\gamma_{0}^{T} &= CC^{-1}\gamma_{\mu}C\gamma_{0}^{T} \\
&= -C\gamma_{\mu}^{T}\gamma_{0}^{T} \\
&= C\gamma_{0}^{T}\gamma_{\mu}^{T}
\end{align}
Hence substituting back we will get<br>
\begin{align}
(i\gamma_{\mu}\partial_{\mu} - m)\psi_{C} &= C\gamma_{0}^{T}(i\gamma_{\mu}^{T}\partial^{\mu}\psi^* - m\psi^*) \\
&= C\gamma_{0}^{T}[(i\gamma_{\mu}^{T}\partial^{\mu}\psi^* - m\psi^*)^{T}]^{T} \\
&= C\gamma_{0}^{T}(i\psi^{\dagger}\gamma_{\mu}\partial^{\mu} - m\psi^{\dagger})^{T} \\
&= C\gamma_{0}^{T}[(i\bar{\psi}\gamma_{0}\gamma_{\mu}\partial^{\mu} - m\bar{\psi}\gamma_{0})]^{T} \\
\end{align}
Now
\begin{align}
\gamma_{0}\gamma_{\mu}\partial^{\mu} &= (\gamma_{0}\partial^{t} + \gamma_{i}\partial^{i})\gamma_{0}
\end{align}
If we substitute back we will get
\begin{align}
(i\gamma_{\mu}\partial_{\mu} - m)\psi_{C} &= C\gamma_{0}^{T}[\{i\bar{\psi}(\gamma_{0}\partial^{t} + \gamma_{i}\partial^{i}) - m\bar{\psi}\}\gamma_{0}]^T \\
&\neq 0
\end{align}</p> | 4,481 | [
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<p>I would like to know more about <a href="http://en.wikipedia.org/wiki/Ehresmann_connection">Ehresmann connections</a> in vector bundles and how they relate to the electromagnetic field and the electron in quantum mechanics.</p>
<p><strong>Background:</strong> The Schrödinger equation for a free electron is</p>
<p>$$ \frac{(-i\hbar\nabla)^2}{2m} \psi = i\hbar\partial_t \psi $$
</p>
<p>Now, to write down the Schrödinger equation for an electron in an electromagnetic field given by the vector potential $A=(c\phi,\mathbf{A})$, we simply replace the momentum and time operator with the following operators</p>
<p>$$\begin{array}{rcl}
-i\hbar\nabla &\mapsto& D_i = -i\hbar\nabla + e\mathbf{A} \\
i\hbar\partial_t &\mapsto& D_0 = i\hbar\partial_t - e\phi
\end{array}$$
</p>
<p>I have heard that this represents a "covariant derivative", and I would like to know more about that.</p>
<p><strong>My questions</strong>:</p>
<ol>
<li><p>(Delegated to <a href="http://physics.stackexchange.com/questions/768/notation-for-sections-of-vector-bundles">Notation for Sections of Vector Bundles</a>.)</p></li>
<li><p>I have heard that a connection is a "Lie-algebra-valued one-form". How can I visualize that? Why does it take values in the Lie-algebra of $U(1)$?</p></li>
<li><p>Since a connection is a one-form, how can I apply it to a section $\psi$? I mean, a one-form eats vectors, but I have a section here? What is $D_\mu \psi(x^\mu)$, is it a section, too?</p></li>
</ol>
<p>I apologize for my apparent confusion, which is of course the reason for my questions.</p> | 4,482 | [
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<p>What papers/books/reviews can you suggest to learn what <a href="http://en.wikipedia.org/wiki/Renormalization" rel="nofollow">Renormalization</a> "really" is?</p>
<p>Standard QFT textbooks are usually computation-heavy and provide little physical insight in this regard - after my QFT course, I was left with the impression that Renormalization is just a technical, somewhat arbitrary trick (justified by experience) to get rid of divergences. However, the appearance of Renormalization in other fields of physics <a href="http://en.wikipedia.org/wiki/Renormalization_group" rel="nofollow">Renormalization Group</a> approach in statistical physics etc.), where its necessity and effectiveness have, more or less, clear physical meaning, suggests a general concept beyond the mere "shut up and calculate" ad-hoc gadget it is served as in usual QFT courses. </p>
<p>I'm especially interested in texts providing some unifying insight about renormalization in QFT, statistical physics or pure mathematics.</p> | 766 | [
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<p>It is well know that, using position representation</p>
<p>$$\langle r\lvert L\rvert \psi\rangle =r \times (-i\hbar\nabla\langle r|\psi\rangle )=r \times (-i\hbar\nabla\psi(r)).$$</p>
<p>However, I read from some books that if $L$ is acting on some position ket directly, then</p>
<p>$$L|r\rangle ~=~ r \times (i\hbar\nabla|r\rangle).$$ </p>
<p>Can anyone explain the latter equation regarding the missing "-" sign?</p> | 4,483 | [
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<p>Suppose that I have a point mass attached to a massless string and I am rotating it vertically. That means The mass is in uniform circular motion and the path of its motion is vertical circle. How does the tension change with respect to the position of the mass. More specifically is the tension in the string is only due to circular motion ($mv^2/R$) or gravity plays a part in it ($mv^2/R$ + something due to weight)?</p> | 4,484 | [
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<p>Lately in my physics class (2nd-year undergraduate) we've been learning about the impulse response of systems and using Green's Functions to model the response of a system to more complicated input forces. Since it's a physics class, not a math class, some of the arguments for the math have been a bit handwavy. I've been going back through the work, then, to try to convince myself of all of the steps. I'm on board with everything up until the final point:</p>
<p>To get the response of a system to a complicated input force, you can find the response of the system to a series of impulses, then add together the resulting responses to get the total response. This requires the assumption that the motion of a system as a function of the given force is linear, i.e.:</p>
<p>$$y(F_1(t))+y(F_2(t))=y((F_1+F_2)(t))$$</p>
<p>Where $y$ is the system's position at some given time, and $F_1(t)$ and $F_2(t)$ are different time-dependent forces applied to the system. More visually, you need the assumption that, in the following picture, it's legitimate to get the actual motion of the system by adding together the two dashed red lines, each of which is the response to an impulse at a different time.</p>
<p><img src="http://i.imgur.com/U3aDbs3.png" alt="Image"></p>
<p>Now, this is obviously correct. The trouble is, I can't see <em>why</em>, mathematically speaking, it should be. My question, then, is how we actually prove that this is the case. I'm sure it's a simple observation that I'm just missing, but I can't for the life of me figure out right now.</p> | 4,485 | [
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<p>Is there any good definition of <a href="http://www.google.com/#q=many+body+localization" rel="nofollow">many body localization</a>?</p>
<p>It is the property of one state or it is the property of a Hamiltonian?</p>
<p>Why does disorder play an important role in many body localization?</p>
<p>What is the relation between Anderson localization and many body localization?</p> | 4,486 | [
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<p>Given what we know about space, time and the movement of galaxies, have we or can we determine what our position is in relation to the projected location of the Big Bang? I've read some introductory papers on the superstructure and galaxy cluster movements, but none of them specifically mentioned space in terms of relative or absolute positions relating to the original position of the Big Bang.</p>
<p>So my question is, does our current understanding of the structure and workings of the Universe give us a good enough estimate to determine our location relative to the Big Bang or can we never guess at it, since every viewpoint in our universe looks the same in every direction?</p> | 4,487 | [
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<p>My (just completed) PhD involved a considerable amount of research involved with the detection of solar UV radiation. This generated quite a bit of interest, especially when I was conducting my experiments outside.</p>
<p>A friend's 6 year old was most fascinated, but could not grasp the concept of UV radiation, primarily as she could not 'see' it. She understood the importance of protecting oneself from too much UV (this is a big thing in Australian schools).</p>
<p>On reflection, it occurred to me that a lot of people don't truly grasp what UV radiation actually is, despite knowing of the risks involved with overexposure. Conversely, I know and understand the technical side of it, but struggle to put it in simpler terms.</p>
<p>What is a simple and meaningful way of explaining UV radiation?</p> | 4,488 | [
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<p>I know that the quantum circuit $\text{CNOT}\; (H \otimes I)$, where $\text{CNOT}$ is the controlled-not gate and $H$ the Hadamard gate, takes the computational basis of two qubits $|00\rangle,|01\rangle,|10\rangle,|11\rangle$ to the Bell states, which are maximally entangled.</p>
<p>Would $\text{CNOT}\;(H \otimes I)$ give us an entangled state even if the qubits are initially not in a standard basis state, <em>i.e.</em> $|0\rangle$ or $|1\rangle$?</p>
<p>I was interested if, when you say that some qubit A is maximally entangled with another qubit B, is there a similar circuit that <strong>(a)</strong> produces that state, and <strong>(b)</strong> from which you can easily show that the qubits are maximally entangled? </p> | 4,489 | [
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] |
<p>Is there a simple way to understand how scientists estimated/calculated the following percentages?</p>
<p><img src="http://i.stack.imgur.com/0GktV.jpg" alt="enter image description here"></p> | 4,490 | [
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<p>Prior to the Dirac delta function, what other distributions functions where physicists using? I find it hard to motivate the theory of generalized functions with just the delta function alone.</p> | 4,491 | [
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<p>I'm studying the diagrammatics for a Bose system (in the superfluid phase) developed by Gavoret and Nozieres (Annals of Physics 28 349 (1964)).</p>
<p>In this paper, they show how to solve the problem using skeleton diagrams. In other words, they give equations for the two-point, three-point and four-point functions, involving the full Green's functions and irreducible diagrams (that they don't discuss, but I think they mean 2-PI irreducible here). In particular, they discuss in great length how these different functions are linked together, and what kind of Ward identities they have to fulfill in order to respect conservation laws.</p>
<p>Of course, this approach is useless unless one does approximations in order to compute correlation functions. What I don't get is how to approximate this skeleton diagrams in a consistent way to recover (for example) standard perturbation theory, which in this context is Bogoliubov theory, and at the same time be sure that the conservation laws are consistently recovered.</p>
<p>I can't find a nice reference that would tell me how to start from these diagrams and what I should do to them. All the textbooks I've looked at only have few pages on skeleton diagrams, and just show how to express the self energies and vertices with them, without discussing anything more.</p> | 4,492 | [
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<p>I have seen the charged pion decay $$\pi^{-}~\to~ \bar{\nu}_{\ell} +\ell^{-}$$ represented with diagrams containing a $W^-$ in the $s$-channel. The $\pi^-$ and $W^-$ have angular momentum $0$ and $1$ respectively, though. How does this process conserve angular momentum?</p>
<p>I see that this question has been asked before, but I haven't found an answers.</p> | 4,493 | [
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<p>To simulate a puck's movement, I use the following model (please restrain from discussing the model, it makes the simulation look the way I want it to) and would like to know how to calculate the total way the puck will move and how long it will take to do so.</p>
<p>My setup is: Per second, the puck looses 20% (<strong>=1-k</strong>) of its previous speed, so after a time <strong>t</strong> the speed will be <strong>v(t) = v(0) * k^t</strong>.</p>
<ul>
<li>If I look now at time <strong>t(n)</strong>, how far has to puck traveled?</li>
<li>How long will it travel until its speed falls below a defined <strong>v(min)</strong>?</li>
<li>How far will it go until its speed drops below <strong>v(min)</strong>?</li>
</ul>
<p>I have tried all kinds of formulas I remember from my physics lessons but I'm really stuck :-(</p> | 4,494 | [
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<p>Suppose we have a mass attached to the top of an ideal (linear and massless) spring oriented vertically in a uniform gravitational field, and on top of that mass there is another mass resting on it. The two masses are not attached at all, so they will lose contact with each other as the normal force is about to become negative. Also suppose that once the two masses separate and collide again, they undergo perfectly elastic collisions.</p>
<p>First of all, is there a name for systems like this? It seems like an "ideal trampoline" to me but searching for that doesn't yield much. Has anyone ever discussed it in a book?</p>
<p>Second of all, is this system chaotic? For sufficiently small oscillations, of course, the masses remain in contact the whole time and you get simple harmonic oscillation, but above some threshold the free mass will keep bouncing off the spring-attached mass and it's quite nontrivial to figure out what eventually happens. Do you get interesting things like period doubling?</p> | 4,495 | [
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] |
<blockquote>
<p><strong>Possible Duplicate:</strong><br>
<a href="http://physics.stackexchange.com/questions/679/learning-physics-online">Learning physics online?</a> </p>
</blockquote>
<p>I'm a high school student, and I got fives in AP Calculus, Mechanics and Electricity and Magnetism exams, and I've taken Linear Algebra and Differential Equations at a local college, and I'll have multivariate calculus done by the end of the year. I was wondering which technique would be more suitable (or important) to learn first, to be able to understand upper-level physics more. Also, does anyone have any recommended texts for those topics?</p> | 145 | [
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<p>What is the definition of a <a href="https://www.google.com/#q=%22charge-neutral+operator%22" rel="nofollow">charge-neutral operator</a>? I guess it means something like: it is invariant under charge conjugation. It that correct? </p> | 4,496 | [
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<blockquote>
<p><strong>Possible Duplicate:</strong><br>
<a href="http://physics.stackexchange.com/questions/17650/how-to-modify-the-bullet-trajectory-based-on-the-ballistic-coefficient">How to modify the bullet trajectory based on the ballistic coefficient?</a> </p>
</blockquote>
<p>I am new to the physics surrounding bullet trajectory and how it is calculated. I am a software developer and I am working on a ballistics calculator. <a href="http://en.wikipedia.org/wiki/Trajectory_of_a_projectile" rel="nofollow">I am using wiki for the trajectory calculation.</a></p>
<p>I am currently using the equation under the "Angle $\theta$ required to hit coordinate $(x,y)$" section. This is all well and good, but it doesn't take into account the drag of the bullet(ballistic coefficient).</p>
<p>I have searched all over trying to figure out how to apply the coefficient to this equation. I am really at a loss and and would be very thankful for any direction in this matter. Maybe I have a gap in my understanding, but I have found plenty of other calculators and other documentation on trajectory and the coefficient but nothing that marries to the two together.</p> | 146 | [
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<p>I imagine that the first ball would strike the rest as normal, but what would the last ball do, without gravity to swing it back? </p> | 4,497 | [
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<p><strong>Does Nantenna (nano antenna) violates 2nd Law of Thermodynamics ?</strong></p>
<p>Nantennas absorb infrared heat and convert it in direct current.
Quote from Wikipedia:</p>
<blockquote>
<p><a href="http://en.wikipedia.org/wiki/Nantenna#Future_research_and_goals" rel="nofollow">He did not discuss whether or not this would violate the second law of thermodynamics.</a></p>
</blockquote>
<p>Some people said that that Nantennas and Solar Cells are not thermodynamics.
I do not understand that thesis. </p>
<p>That is why I have re-phrased question:</p>
<p><strong>Are Nantennas a perpetual motion machine of Second Kind ?</strong></p>
<p>As I know: 2nd law XOR perpetual motion of second kind</p>
<p><strong>PS: I can't agree that answers are prooving that that second law is not violated by nantennas and MIM diods. If someone has to say anything about the subject then I would be happy to read that. Thank You all for responses!</strong></p> | 4,498 | [
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<p>In conventional conductors, the RC time constant is the time required to charge or discharge a capacitor through a resistor by ≈ 63.2 percent of the difference between the initial value and final value:</p>
<p>$$\tau = R \cdot C $$</p>
<p>However, in a superconductor, the resistance is exactly zero. </p>
<p>$$\tau = 0 \cdot C = 0$$</p>
<p>Which would require infinite current because the capacitor is charged in 0 time. So the RC time constant equation above must not be valid for superconductors. How should the RC time constant be defined in a superconductor such that it does not require infinite current?</p>
<p>That is, what is the expression for the time constant of charging a superconducting capacitor by ≈ 63.2 percent?</p>
<p><strong>Edit: I re-framed the question because we were getting bogged down about signal propagation time. Now instead of using special relativity to point out the flaw in the equation, I note that it would require infinite current, which would not be physically realizable.</strong></p> | 4,499 | [
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<p>I think Physics is fascinating , especially those really clever experiments one can do that demonstrate some important principle and probably don't cost a lot of money to make. Also the Thought Experiments of Galileo and Einstein and Maxwell (I think) and many others ; thought experiments that cost nothing and point out interesting ideas. It's great when some scientist figures out a relatively inexpensive way to test some complicated idea. I know there is a great 'need' to analyse the fundamental structures of matter but could this be done without spending a million dollars a day as they do with the Large Hadron Collider( If that monetary value is accurate). Can expensive experiments be redesigned so they are inexpensive yet still effective?</p> | 4,500 | [
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<p>What would be the requirement to learn matrix mechanics?</p>
<p>More specifically, what math do I need?</p>
<p>Can anyone recommend me a book that covers all maths needed for matrix mechanics?</p> | 4,501 | [
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<p>If the Higgs field gives mass to particles, and the Higgs boson itself has mass, does this mean there is some kind of self-interaction?</p>
<p>Also, does the Higgs Boson have zero rest mass and so move at light-speed?</p> | 269 | [
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<p>Could an 18th century or earlier scientist have come across phenomena which require quantum theories to explain them, given the apparatus available at the time?</p>
<p>I'll choose 1805 as the cut-off date, because that's when Maudslay's micrometer revolutionised precision in instruments.</p> | 4,502 | [
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<p>There is an event horizon where cosmic expansion leads to superluminal recession speeds for sufficiently distant objects -- the Hubble Volume.</p>
<p>1) Does matter beyond the event horizon affect us gravitationally or otherwise?</p>
<p>If the answer is no, there is a follow-up question.</p>
<p>If everything beyond this horizon is causally disconnected, it gives rise to the possibility that the universe is arbitrarily large, and <strong>undetectably</strong> so...</p>
<p>If the distribution of matter is random then although it is locally smooth, in a sufficiently large universe there may be large regions on the thin tail of the bell curve which are relatively empty or relatively full...</p>
<p>We've all seen that chart of possible Hubble constants: <1 means big crunch, >1 means heat death, =1 means asymptotic growth. We've all seen that the constant appears to be very, very close to 1...</p>
<p>Putting all this together...</p>
<p>2) Could we have a situation in which two relatively dense regions of the universe, separated by a sparse region, expand away from one another faster than the average rate due to weaker attraction locally, and so end up causally separated beyond the event horizon of expansion, while within each region there is a local big crunch?</p> | 4,503 | [
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<p>I was thinking; <em>what shape does a black hole have?</em>. By 'Shape', I mean its form (e.g, circle , cylinder, sphere, torus, etc..).</p>
<p>We usually think of black holes as if they're plugholes (e.g, a flat circular object), but what if they're spherical? A spherical black hole would make much more sense.
I would imagine than a black hole shaped like a basketball would be capable of pulling more mass towards it than a flat one, as it has a higher surface-to-volume ratio to do so.</p>
<p><em><strong>Edit</em></strong>
I know that it's probably a sphere, but when you think about it, a cylinder could also be a potential shape.</p> | 4,504 | [
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<p><br>Due to an experiment, I need a small heater (around 70 to 100 watts). I intend to use an incandescent bulb so it can act as a heater. What I wonder here is will a 70 watts Incandescent Bulb be equal to a 70 watts heater? </p> | 4,505 | [
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<p>If the electromagnetic field of an unpolarized plane wave is written as
$$\bar{E}(t,\bar{x})=(\bar{E}_{0x}+\bar{E}_{0y}e^{i\delta(t)})e^{i(\bar{k}\bar{x}-\omega t)}$$
$$\bar{B}(t,\bar{x})=\frac{1}{\omega}\bar{k}\times\bar{E}(t,\bar{x})$$
where $\delta(t)$ is a random phase shift, then the intensity of this light is given by the time-average of the norm of the Poynting vector
$$I(\bar{x})=\left<\|\bar{P}(t,\bar{x})\|\right>_{t}$$
$$\begin{split}\bar{P}(t,\bar{x})=&\frac{1}{\mu_{0}}\mathcal{R}e(\bar{E}(t,\bar{x}))\times\mathcal{R}e(\bar{B}(t,\bar{x}))\\
=&\frac{1}{\omega}\bar{E}_{0x}^{2}\cos^{2}(\bar{k}\bar{x}-\omega t)\bar{k}+\frac{1}{\omega}\bar{E}_{0y}^{2}\cos^{2}(\bar{k}\bar{x}-\omega t+\delta(t))\bar{k}
\end{split}$$
$$\Leftrightarrow I(\bar{x})=\frac{1}{2}c\epsilon_{0}\bar{E}_{0x}^{2}+c\epsilon_{0}\bar{E}_{0y}^{2}\left<\cos^{2}(\bar{k}\bar{x}-\omega t+\delta(t))\right>_{t}$$
Can we simplify this further? Is the remaining average also $1/2$?</p> | 4,506 | [
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<p>I haven't taken Physics in University. Lately, I've been reading about some of the branches of physics through Wikipedia. I read several times that many of the theoretical models do not explain why time only moves forward. And that the theoretical models support the ability for time to move backwards. </p>
<p>I'm having difficulty understanding what happens to human consciousness when time moves backwards. I can "perceive" time going forwards. I am learning something every day, new neural connections are forming every moment in my brain. If time moves backward, do I "forget" what I've learned because my neural connections are deconstructed? All the electro-chemical reactions in my brain/body will proceed in opposite direction? </p>
<p>I can't tell if I'm asking a philosophical, biological or physics question...</p>
<p><strong>Addition</strong> — Maybe part of the answer I'm looking for is whether the universe is deterministic going forwards in time and backwards in time. E.g. Just because I accidentally spill a cup of milk, when time is moving forwards doesn't necessarily mean that same milk will pour back into the cup when time moves backwards?</p> | 4,507 | [
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<p>I want to experiment with an enclosure for my phone so the frequency response has a little more punch at the bottom end. I understand that something can't be created from nothing, but enclosures work for drivers so I can't see why not for the phone?</p>
<p>What sort of cabinet design would do the job? It would be nice if it preserved a natural mid and treble as well. Even if this design just dampens medium and high frequencies that would be fine too. The response without any enclosure seems to have started to tail off at about 260Hz and is almost gone by 130Hz. I can just about hear 60Hz if I put the volume on full and my ear against the speaker. (Don't try this on your hifi at home kids)</p>
<p>Thanks</p> | 4,508 | [
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<p>One of the most accepted frameworks for the relationship between the magnitude and frequency of an earthquake is that of the critical phenomena. In this framework, the magnitude of events must be distributed following a power law. </p>
<p>However the <a href="http://en.wikipedia.org/wiki/Gutenberg%E2%80%93Richter_law" rel="nofollow">Gutenberg-Richter law</a> clearly shows a deviation at low magnitudes from the power law called a <em>roll-off</em>. How can the roll-off deviation of the GR prediction be explained?</p> | 4,509 | [
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<p>I read here (<a href="http://en.wikipedia.org/wiki/Proton#Quarks_and_the_mass_of_the_proton">mass of a proton</a>) that the mass of a proton is mostly (99%) due to the energy of the strong nuclear force which binds the quarks together, and not the actual mass of the quarks. My question is: if the quarks didn't have <em>any</em> mass, would an arrangement like the proton even be possible? Could mass-less quark-like particles combine to form a massive proton-like thing?</p> | 4,510 | [
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<p>I was reading a paper dealing with the Hilbert of quantum gravity (or more precisely what should it look like considering what we know from QM and GR) ref: <a href="http://arxiv.org/abs/1205.2675" rel="nofollow">http://arxiv.org/abs/1205.2675</a> and the author writes the following:
$${\cal{H}_M} = {\cal{H_{M,\,\textrm{bulk}}}}\otimes{\cal{H_{M,\,\textrm{horizon}}}}$$ for a specific manifold $\cal{M}$. I know very little about the holographic principle and the AdS-CFT correspondence but isn't it a redundant description? If there is a duality between the gravitational theory in the bulk and the CFT on the boundary, knowing one means knowing the other, so why can't we restrict ourselves to one of the Hilbert spaces? Moreover, the author writes, a couple of lines after this first element, that the two Hilbert space have same dimension ( $\textrm{exp}({\frac{{\cal{Area}}}{4}})$ ) so they are totally equivalent, as a complex Hilbert space is only defined by its dimension.</p> | 4,511 | [
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<p>We know that the <a href="http://en.wikipedia.org/wiki/Sound" rel="nofollow">sound</a> waves propagate through air, and it can't travel through vacuum. so the thing that help it doing that is the air's molecules pressure. So my question how can that happens? I can't understand that concept.</p> | 4,512 | [
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<p>What actually is 1 light year? What is the equivalent time in Earth time space?</p> | 4,513 | [
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<blockquote>
<p><strong>Possible Duplicate:</strong><br>
<a href="http://physics.stackexchange.com/questions/47266/rolling-without-slipping-ball-on-a-moving-surface">Rolling (without slipping) ball on a moving surface</a> </p>
</blockquote>
<p>Apparently I didn't log in properly when I asked a question this morning: <a href="http://physics.stackexchange.com/questions/47266/rolling-without-slipping-ball-on-a-moving-surface">Rolling (without slipping) ball on a moving surface</a> and now I couldn't go back to leave comments. I am starting over here, so apologies for the confusion and inconvenience.</p>
<p>The user Raindrop left an answer to the above-mentioned question and I think he pointed out where I got confused. In the case where the surface (I am imagining more of a 2D arc swinging from side to side) is moving as the ball (appear as a simple circle) is rolling on it, the sum of force is not simply $F=ma=mgsinθ−f$, where $f$ is the frictional force acting on the ball. It was pointed out that I am missing the normal force. So here's my question: what is this normal force? I thought the normal force is what the surface exerts on the ball due to the gravitational force the ball is exerting on the surface. I thought the normal component of that force would be cancelled, leaving the $mgsinθ$ component only. Could someone explain that to me please? Much appreciated.</p> | 147 | [
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<p>Reading this PE question
<a href="http://physics.stackexchange.com/questions/5702/can-we-transport-energy-over-infinite-distances-through-vacuum-using-light">can-we-transport-energy-over-infinite-distances-through-vacuum-using-light</a>, a related question arises naturally: </p>
<p>Is <strong>energy transported</strong> (by light)? -- (I did believed in this answer until now)
or <strong>energy is already 'in site'</strong> (vacuum) just expecting to be excited by the photons? </p>
<p>This news insinuated the doubt <a href="http://www.wired.com/wiredscience/2011/02/real-live-antilaser/" rel="nofollow">anti-laser</a>(1)</p>
<blockquote>
<p>The antilaser does the reverse: <strong>Two
perfect beams of laser light go in,
and are completely absorbed.</strong></p>
</blockquote>
<p>If vacuum is able to absorb energy then it can do the reverse, and supply energy.
We are already prepared to accept that <a href="http://en.wikipedia.org/wiki/Vacuum_energy" rel="nofollow">vacuum has energy</a>. </p>
<p>I am inclined to accept that <strong>energy do not travel at all</strong>. What is travelling is the excitation of vacuum, and we call this: <em>photons</em>.
It may appear a question of semantics, but I think that the explicit reconaissance of this notion can be helpful.</p>
<p>(1) The two rays entering the slab are in phase opposition when they met, and cancel. The nature of cancelation was obscure to me, until now. </p>
<p><strong>added:</strong></p>
<p>I googled this: "where goes the energy in a destructive interference" and followed past answers to this question. Someone answered "into the surrounding environment." We are in minority;) Most of the times they said that the total extinguishing is impossible. This anti-laser experiment shows that energy is destroyed.</p>
<p>We see the same effect with sound cancelation , with boat wake (trailing waves) cancelation (by double/triple hull or when they sail in formation), and now with light.</p>
<p><strong>added after 2 answers</strong><br>
<img src="http://i.stack.imgur.com/xrA2J.jpg" alt="Destructive interference"> image from <a href="http://www.astro-canada.ca/_en/a2313.html" rel="nofollow">astro-canada.ca</a><br>
What is amazing is that this fact is inside the theory since the begining, quoting from there:</p>
<blockquote>
<p>In 1801, the British physicist Thomas
Young demonstrated that light
propagates as waves, like waves on the
surface of water. Young understood
that when two light waves meet, they
interact with each other. Scientists
call this “interference”.
In the opposite scenario, where the
crests of one wave are aligned with
the troughs of another, they cancel
each other out and the <strong>light
disappears</strong>. This is destructive
interference.</p>
</blockquote>
<p>As anna says in her answer the actual theorethical framework does not understand a vanishing of energy.
The question remains open: What happens to energy when '<strong><em>light disappears</em></strong>' or '<em>light cancels</em>' or '<em>destructive interference</em>'? </p> | 4,514 | [
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<p>This is a spin-off of the following question: <a href="http://physics.stackexchange.com/q/98001/">Are Thomas Breuer's subjective decoherence and Scott Aaronson's freebits with knightian freedom the same things in essence?</a></p>
<p>Given that Thomas Breuer has proven that universally valid theories either deterministic or probabilistic are impossible, I wonder whether the result extends to possibilistic (rather than probabiblistic) theories as well.</p>
<p>By possibilistic physical theories I mean such theories in which the most complete physical description of a system includes uncertain probability. For instance, those who employ the mathematical apparatus of Dempster-Shafer theory, Graded possibilities, Sugeno Lambda-measures, Belief and Plausibility measures, Fuzzy set theory and
others studies under recently-developed <a href="http://worldtracker.org/media/library/Science/Probability%20&%20Statistics/Uncertainty%20And%20Information%20-%20Foundations%20Of%20Generalized%20Information%20Theor;%20y%20George%20J.%20Klir%20%28Wiley,%202006%29.pdf" rel="nofollow">Generalized Information Theory (GIT)</a></p>
<p>Can such theory be universally valid in absolute (rather than relative) sense (albeit having even less predictive power than a probabilistic theory)?</p> | 4,515 | [
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<p>I was reading Stephen Hawking's 'The theory of everything' when I came across a very interesting type of universe, the 'zero energy universe' since then, I've read some websites but all they used to prove the hypothesis's authenticity were observations(which was found to be very close to zero, and there would anyways be an unexplained curvature if the energies were not close to zero) </p>
<p>What I am interested in knowing is that in the book, Stephen hawking said, and I am paraphrasing that we can add as much mass as we want to the universe because subsequent negative gravitational potential energy will get added.</p>
<p>Now the boldness with which the line was written, I am convinced that there must be some mathematical proof of this relation between the mass and negative energy being exactly equal to each other, at least in a universe without complications like quantum effects. However, I was not able to find any such proof. Can the universe cook books?</p> | 4,516 | [
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<p>I am trying to find a solid material that almost fully (since there is not a thing that can fully insulate electricity) blocks static charges from one layer to another. I know plastic is a good insulator but I want to know if there is a better insulator than plastic. I am going to use this material as a clothing for a project so the material shouldn't be easily breakable like glass.</p> | 4,517 | [
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<p>If there are no localized observables in quantum gravity, does spacetime really exist, or might spacetime really be an illusion?</p> | 4,518 | [
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<p>In a previous question on <a href="http://physics.stackexchange.com/questions/3773/shape-of-the-higgs-branching-ratio-to-zz">Higgs branching ratios</a>, I find this image</p>
<p><img src="http://i.stack.imgur.com/eljaD.jpg" alt="enter image description here"></p>
<p>(originally from page 15 <a href="http://arxiv.org/abs/hep-ph/9704448">here</a>). </p>
<p>I am VERY intrigued by the fact that decays to WW, gg, and ττ are almost equally probable, for a standard model Higgs with a mass in the vicinity of 115 GeV. I've <a href="http://physics.stackexchange.com/questions/8999/115-gev-170-gev-and-the-noncommutative-standard-model">noted previously</a> that this is a special value: it lies in a narrow range of values for which the SM vacuum is metastable. Is there a connection? </p> | 4,519 | [
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<p>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?</p> | 4,520 | [
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<p>let's say we pour some water to the Sun. Or boil water on 2000 degrees Celsius on Earth.</p> | 4,521 | [
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<p><strong>Problem/Purpose of me asking this question to you people who know more than me:</strong></p>
<p>So I'm doing a science project where I'm collimating a beam of light to a focus point in a light medium (water vapor or fog) and I want to calculate the intensity near it. I can't seem to find an equation that describes this problem. I want to know two things. If you know anything that can be related in solving this problem, it is much appreciated! :)</p>
<p><strong>Issues/Things I need help to figure out!:</strong></p>
<p>(1) If I focus a collimating beam of light with a lens, (say a hand held magnifying glass), into a relatively uniformly dispersed light medium, (water vapor or theatrical fog) Can this focus point be seen in <strong>ANY direction</strong>, (say like 5 feet away from the focus point)? Doesn't light scatter isotropically in this case? IF NOT, What is the preferred direction of scattering of the light?</p>
<p>(2) If an equation exists (And light scatters isotropically in the light medium used), can this equation say given the parameters, (light frequency used, index of refraction of medium, density of medium, size of collimated beam, lens dimensions used, etc) give the intensity of light in terms of the distance a person is away from the focal point? I am aware of the <strong>inverse square law</strong>, but in my case, its a bit different, isn't it?</p>
<p>Wouldn't my situation involve some type of directionality? How do you find the best viewpoints from the focal point in which the intensity of the focal point is most profound?</p>
<p><strong>MORE relevant or related questions that need to be addressed:</strong></p>
<ul>
<li>Does the particle size matter?, (the particles that make up the light medium)</li>
<li>How do I determine the correct density of the given medium to produce the most profound effect, (Having the focal point illuminate as brightly as possible)</li>
<li>How do I determine the right intensity of the initial column of light that is focused?</li>
<li>Combining the previous two bullets, How do I determine the right combination of the density of light medium and intensity of light used to illuminate the focus point as bright as possible?</li>
</ul>
<p><strong>The Gist:</strong></p>
<p>What I'm trying to do is to create a "point of light" inside a suspended light medium (Ideally viewable in all directions) and I'm trying to figure a way to figure this out with equations before buying a whole ensemble of things (Fog machine, light source/laser, magnifying glass or multiple lenses, etc) to test it. (If not viewable in all directions, and it has directionality, then I'm just gonna combine multiple systems pointing multiple columns of focused light in different directions to the same point in space to obtain an acceptable looking "point of light" in a medium).</p> | 4,522 | [
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] |
<p>In the renormalization procedure, is writing things like</p>
<p>$$\varphi=\sqrt{Z_{\varphi}}\ \varphi_R\ ,\ \ m_0^2=Z_m\ m_R^2\ ,\ \ g_0=Z_g \mu^{\epsilon}\ g_R$$
and
$$Z_i=1+\sum_{\nu=1}^\infty C_i^{(\nu)}(m_R,\mu,\Lambda\text{ or }\epsilon)·g_R^\nu\ , \ \ \ \ \ i=\varphi, m, g$$
really more than just an arbitrary ansatz?</p>
<p>I have no idea what principle people follow, when people have a Lagrangian, say for QED and then write down Lagrangians in the to-be-renormalized-stage. There seems to be a motivation to make them look similar to the old Lagrangian before introducing that coupling constrant expansion - and why in $g$, not other variables like $m$? Hence they write things like $m_{old}=c·m_{new}$, which seems faily conservative, because it doesn't introduce new terms, beyond maybe counter terms that look structurally list the old ones. But as far as I can see, the theory really just starts with the Lagrangian, which contains the to be found $Z$-expressions. You don't use the Lagrangian before that, do you? At least not beyond tree graphs. Therefore I think you could just begin with a buch of terms, with object that have to be fitted by renormalization. The theory effectively seems just to start with the non-bare object.</p>
<p>From all the possible 'unphysical numbers' in the expansion for the (finite number of) $Z$-terms, why does only the 'scale' $\mu$ survive? Do all scheme leave one number open, and if yes, why? I don't get the what this object '$\mu$' is, at all.</p> | 4,523 | [
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<p>Some equipment sometimes have a high pitch ringing, and I was wondering out of curiosity: can noises (not only drive you crazy but also) settle and become permanent ringing in your ear if you are exposed long enough?</p> | 4,524 | [
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<p>Few days ago I started making physics engine on directx. As it obvious I have encountered one problem. I can't find formula of air resistance for angular velocity. I only able to find drag force, but as I suspect it only works for linear velocity(<a href="http://en.wikipedia.org/wiki/Drag_(physics)" rel="nofollow">http://en.wikipedia.org/wiki/Drag_(physics)</a>) :). So, please help me! :D</p>
<p>By the way, I saw one post with sphere air resistance angular velocity <a href="http://physics.stackexchange.com/que...-ball-in-fluid">http://physics.stackexchange.com/que...-ball-in-fluid</a>, but I need it with box. In addition, for angular velocity I'm using this formula:</p>
<blockquote>
<p>F = 0.5 c * ρ * A * v^2</p>
</blockquote>
<p><code>F</code> - is the drag force,<br>
<code>ρ</code> - is the density of the fluid,<br>
<code>v</code> - is the speed of the object relative to the fluid,<br>
<code>A</code> - is the cross-sectional area, and<br>
<code>c</code> - is the drag coefficient – a dimensionless number.</p> | 4,525 | [
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<p>In the plasma literature, noble gases—usually helium or argon—are frequently said to 'stabilize' plasmas. For instance in <a href="http://patentimages.storage.googleapis.com/pdfs/US5454903.pdf" rel="nofollow">this patent</a>, the inventor states that the plasma can be stabilized "by adding an electron-donor gas (i.e. a species having a low ionization energy)".</p>
<p>I understand how this would help by making more electrons available to sustain the plasma, especially in the presence of an electron-hungry compound like NF₃, but <strong>I don't understand why helium is considered a good electron donor</strong> when the Wikipedia article puts helium at the very top of the elements (~25eV) for ionization energy. In fact, <a href="http://dx.doi.org/10.1016/j.ijms.2012.05.005" rel="nofollow">this article</a> calculates the first ionization energy of NF₃ to NF₃+ to be 13.5eV, and other pages list the ionization of N₂ and H₂ to be ~1500kJ/mol (~15.5eV).</p>
<p>(I'm especially confused because I've seen helium stabilize an atmospheric pressure plasma in person: my collaborator's dielectric barrier discharge He:O₂ plasma jet won't ignite if it's run with pure O₂ or O₂+air, but it works perfectly when He is added.)</p> | 4,526 | [
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<p>I understand that the <a href="http://en.wikipedia.org/wiki/Dirac_equation" rel="nofollow">Dirac equation</a> has negative and positive sets of solutions and this contributes to its quantization by a superposition of two Fourier modes represented as creation and annihilation operators. What about a complex Dirac field for representing antiparticle fields?</p>
<p>I don't understand why a real field can't describe the antiparticles alone since it was the negative solutions of the real Dirac field which first sparked the antiparticle debate.</p> | 4,527 | [
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<p>Who "invented" the concept of symmetries? This article is quite extensive, but it blurs the history with the modern understanding.
<a href="http://plato.stanford.edu/entries/symmetry-breaking/" rel="nofollow">http://plato.stanford.edu/entries/symmetry-breaking/</a></p>
<p>Some of the concepts can be traced to Galileo and Newton, but I'm quite certain the modern notion is incompatible with their view of the world. Does the notion come from group theory specifically? Can the first mention be traced accurately?</p>
<blockquote>
<p>Although the spatial and temporal invariance of mechanical laws was
known and used for a long time in physics, and the group of the global
spacetime symmetries for electrodynamics was completely derived by H.
Poincaré [7] before Einstein's famous 1905 paper setting out his
special theory of relativity, it was not until this work by Einstein
that the status of symmetries with respect to the laws was reversed.</p>
</blockquote> | 4,528 | [
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<blockquote>
<p><strong>Possible Duplicate:</strong><br>
<a href="http://physics.stackexchange.com/questions/30268/basic-buoyancy-question-man-in-a-boat-with-a-stone">Basic buoyancy question: Man in a boat with a stone</a> </p>
</blockquote>
<p>If you have a large boulder on a boat, in a pond, and you throw the boulder overboard and into the pond, would the water level decrease, increase, or remain the same?</p>
<p>I believe that the level of water would remain constant because the same amount of force is acting on the water regardless of the rocks position, thus the system is displacing the same amount of water. </p> | 148 | [
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<blockquote>
<p><strong>Possible Duplicate:</strong><br>
<a href="http://physics.stackexchange.com/questions/45802/stiffness-tensor">Stiffness tensor</a> </p>
</blockquote>
<p>Let's have a stiffness tensor:</p>
<p>$$
a^{ijkl}: a^{ijkl} = a^{jikl} = a^{klij} = a^{ijlk}.
$$</p>
<p>It has a 21 independent components for an anisotropic body. </p>
<p>How does body symmetry (cubic, hexagonal etc.) change the number of independent components of the tensor? For example, for cubiс symmetry it has three components. How to explain it?</p>
<p>Update.</p>
<p>Is the explanation a simple realization of idea
$$
a_{ijkl}' = \beta_{im}a^{m}\beta_{jt}a^{t}\beta_{k f}a^{f}\beta_{ld}a^{d} = a_{ijkl},
$$
where $\beta_{\alpha \beta}$ is a components of a matrix $\beta$ for rotation around z-, x-, y-axis at the same time?</p> | 149 | [
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<p>Why do orbital speeds decrease further away from the focus? A simple question, but I want to make sure I am understanding this correctly: Is it ONLY a function of the gravity well? As in, the gravitational field is weaker as you move away from the massive body, so the speed decreases? What if the gravitational field was constant through space? Would the orbit's speed then be constant?</p>
<p>This should be a home-run for someone. </p> | 4,529 | [
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<p>Suppose I have a wave function $\psi(x)$ in position basis. I can make a density function
by simply multiplying $\psi(x)$ and its conjugate $\psi^*(x)$. If I operate the density matrix $\rho(x,y)=\psi(x)\psi^*(y)$ with Hamiltonian $\hat{H}$ I will get,
\begin{equation}
\hat{H}\rho(x,y)=\hat{H}\psi(x)\psi^*(y)=\psi^*(y)\hat{H}\psi(x)+\psi(x)\hat{H}\psi^*(y)
\end{equation}
since the hamiltonian is a diferential operator.
At the same time, If I take discrete basis representation, I will get $\langle n|\psi\rangle=V_{1}$ and $\langle\psi| n\rangle=V_{1}^{\dagger}$ which are the column vector and row vector respectively. Now if I operate the Hamitonian matrix $\hat{H}$ on the density matrix I will get,
\begin{equation}
\hat{H}\rho=H (V_{1} V_{1}^{\dagger})=(H V_{1}) V_{1}^{\dagger}
\end{equation}
That means, I need to apply the hamiltonian operator only once on a vector. Why it is not distributed like the differential operator.</p>
<p>\begin{equation}
\hat{H}\rho= V_{1} H V_{1}^{\dagger}+(H V_{1}) V_{1}^{\dagger}
\end{equation}
Why this equation is not valid in the case of discrete basis ?</p> | 4,530 | [
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<p>How do I find the most probable value of position of a (non-Gaussian) wave function? Is it the same value as the expectation value of the position? And is it true that the most probable value of position is equal to the mean for a Gaussian? </p> | 4,531 | [
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<p>I'm working on a pre-lab for my Physics 1 lab session, and I had a debate with the person I carpool with (who is taking the algebra-based Physics 1 lab). We seem to be unsure about uncertainties, and how they play out when doing calculations.</p>
<p>The given question was as follows:</p>
<blockquote>
<p>When the falling mass is $0.250\text{ kg}$, a student obtains an acceleration
from part 2.1 of $0.3±0.1\ \mathrm{m/s^2}$ and the radius of the shaft from part
2.2.1 of $0.015\text{ m}$.</p>
<p>a.) What is the expected angular acceleration as for part 2.2.2? The
measured angular acceleration from part 2.4 is $16.1±0.3\ \mathrm{rad/s^2}$.</p>
</blockquote>
<p>The given formulas were: </p>
<p><img src="http://i.stack.imgur.com/qFxv5.png" alt="table of given formulas"></p>
<p>I used the acceleration formula, and came up with:</p>
<p>$$\begin{align}
a &= \alpha r_\text{shaft} \\
(0.3\pm0.1) &= \alpha\cdot(0.015) \\
\frac{(0.3\pm0.1)}{(0.015)} &= \alpha \\
13.33\;\mathrm{\frac{rad}{s^{2}}} &\leq \alpha \leq \; 26.67 \;\mathrm{\frac{rad}{s^{2}}} \\
\alpha &= 20.00 \pm 6.67 \;\mathrm{\frac{rad}{s^{2}}}
\end{align}$$</p>
<p>My friend says that is incorrect (although I am fuzzy on his reasoning), but he mainly says you can immediately tell because the uncertainty is so high (30%+ from the value). My argument to that is, the original given uncertainty (±0.1) is 30% of the original value, so why can't the result's uncertainty be 30% of the calculated value?</p>
<p>My friend does make a good point, the uncertainty for the value is very high. But does the large uncertainty in the given acceleration make it okay for the final uncertainty to be that high?</p> | 4,532 | [
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<p>The <a href="http://en.wikipedia.org/wiki/Quantum_eraser_experiment" rel="nofollow">quantum eraser experiment</a> tells us that a photon shot at two slits is a wave, unless you measure which slit is taken <strong>and</strong> you do not destroy the measurement result.</p>
<p>I've found this very similar to the notion of '<a href="http://en.wikipedia.org/wiki/Lazy_evaluation" rel="nofollow">lazy evaluation</a>' in computer science. Only evaluate when it is certain the result is required.</p>
<p>Konrad Zuse's theory for '<a href="http://en.wikipedia.org/wiki/Calculating_Space" rel="nofollow">rechnender raum</a>' or a computing universe was always just a theory.</p>
<p>But could the behaviour of photons not be described as the 'lazy evaluation' of a computed universe. If the 'which slit information' is not required, the universe never bothers to compute this information. Somehow, the computed universe is optimized?</p> | 4,533 | [
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<p>Fitness Model, Rob Riches, claims that doing bicep curls with Olympic bars is different than lifting with normal bars. </p>
<blockquote>
<p>Biceps have always been a favorite muscle group of mine to train, but
since I’ve started using the Olympic bar to curl with, I’m noticing my
arms start to thicken out, where as before they had good size from the
side view, but when viewed from straight on, I felt as though they
lacked the width I should have in them.</p>
</blockquote>
<p><a href="http://www.simplyshredded.com/wbff-profitness-modelbodybuilder-rob-riches-talks-with-simplyshredded-com.html" rel="nofollow">Source</a></p>
<p><strong>Normal barbell curl</strong> (short)</p>
<p><img src="http://i.stack.imgur.com/ZGCUI.jpg" alt="barbell curl"></p>
<p><strong>Olympic barbell curl</strong> (long)</p>
<p><img src="http://i.stack.imgur.com/7yBD3.jpg" alt="olympic barbell curl"></p>
<p>Does Rob Riches' tip make sense when explained with physics? Let's take two barbells. The first barbell is an Olympic barbell loaded to a total of 60 lbs. The second barbell is a normal length barbell at 60 lbs as well. How would the force on the biceps differ when lifting these two different length but identical weight barbells? I don't see how the biceps would feel any difference. </p> | 4,534 | [
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<p>I heard that we must know the Weyl tensor for fully describing the curvature of the 4-dimensional space-time (in space-time with less dimensions it vanishes, so I don't interesting in cases of less dimensions). So I have the question: what is physical (or geometrical) sense of the Weyl tensor and why don't we need only Riemann tensor for describing the curvature? Does it connected with gravitational waves directly?</p> | 4,535 | [
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<p>Ok, so we can have conformal invariance on a string world sheet. However, it is well known that to preserve conformal symmetry we require states to be massless. So how is it that string theories incorporate CFT but allows massive states?</p>
<p>Is it because the CFT is on the worldsheet and therefore applies to the worldsheet coordinate X (X is treated as the field) - however the physical states arise from the the creation/annihilation operators that we get from X? Therefore the CFT doesn't actually act on the states (massive or massless) but instead it acts on the field X. </p> | 4,536 | [
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<p>I'm reading the Wikipedia page for the <a href="http://en.wikipedia.org/wiki/Dirac_equation" rel="nofollow">Dirac equation</a>:</p>
<blockquote>
<p>$\rho=\phi^*\phi\,$</p>
<p>......</p>
<p>$J = -\frac{i\hbar}{2m}(\phi^*\nabla\phi - \phi\nabla\phi^*)$</p>
<p>with the conservation of probability current and density following
from the Schrödinger equation:</p>
<p>$\nabla\cdot J + \frac{\partial\rho}{\partial t} = 0.$</p>
<p>The fact that the density is positive definite and convected according
to this continuity equation, implies that we may integrate the density
over a certain domain and set the total to 1, and this condition will
be maintained by the conservation law. A proper relativistic theory
with a probability density current must also share this feature. Now,
if we wish to maintain the notion of a convected density, then we must
generalize the Schrödinger expression of the density and current so
that the space and time derivatives again enter symmetrically in
relation to the scalar wave function. We are allowed to keep the
Schrödinger expression for the current, but must replace by
probability density by the symmetrically formed expression</p>
<p>$\rho = \frac{i\hbar}{2m}(\psi^*\partial_t\psi - \psi\partial_t\psi^*).$</p>
<p>which now becomes the 4th component of a space-time vector, and the
entire 4-current density has the relativistically covariant expression</p>
<p>$J^\mu = \frac{i\hbar}{2m}(\psi^*\partial^\mu\psi - \psi\partial^\mu\psi^*)$</p>
</blockquote>
<ol>
<li><p>What exactly are $\partial_t$ and $\partial^\mu$? </p></li>
<li><p>Are they tensors? </p></li>
<li><p>If they are, how are they defined?</p></li>
</ol> | 4,537 | [
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<p>The title says it all. If I'm standing in the wind and I'm wet, I feel much
colder than when I'm dry. This is true no matter how warm or cold the water.
Why is this?</p> | 95 | [
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<p>i have two sets of data. one leads to a value for the transition temperature from the A phase superlfuid to B phase. This was performed in a thin slab, which was too thin to observe the A-B transition. I also have an actual value of this temperature from another set of data where the slab was thick enough to support the nucleation of the B phase. I need a way of comparing the two values to see if the extrapolation was correct. How can i relate the temperature to pressure and thickness/reduced thickness/cohenerence length? </p> | 4,538 | [
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<p>In the <a href="http://en.wikipedia.org/wiki/Muon">Wikipedia article of Muon</a>, it says</p>
<blockquote>
<p>...with unitary negative electric charge of <strong>roughly</strong> -1 and a spin of 1/2,</p>
</blockquote>
<p>What are they trying to convey with the "roughly"? Aren't the allowed values of charge discrete?</p> | 4,539 | [
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<p>I am confused about a trivial concept. Let the rotation of a rigid body, say with one point fixed, be described by the equation $\vec{x}(t)=R(t)\vec{x}(0)$, with $R(0)=I$. </p>
<p>Then, at each instant there is only one real eigenvector of $R(t)$ with eigenvalue 1 that we may call $\vec{v}(t)$ and which we may take to be normalized. That vector $\vec{v}(t)$ is what geometrically we would call the (instantaneous) axis of the rotation. </p>
<p>Kinematically, however, the instantaneous axis of rotation is the line of points with vanishing instantaneous velocity $\dot{\vec{x}}(t)=\vec{\omega}(t)\times\vec{x}(t)=0$. That is the direction of $\vec{\omega}(t)$.</p>
<p>As is obvious (for example from the Rodrigues formula), in general $\vec{v}(t)$ and $\vec{\omega}(t)$ are not parallel. So, why are there two axes of rotation, and does $\vec{v}(t)$ play any role in the kinematics/dynamics of the motion?</p> | 4,540 | [
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<p>I know that the nuclear force is responsible for binding the protons and neutrons together in the nucleus. The force is powerfully attractive at small separations and rapidly decreases as the distance between the particles concerned increases and becomes repulsive after that.But, why does that happen? </p>
<p>I'm not able to find a way to explain it in anyway.How can a force be attractive and repulsive based on the difference between the concerned particles? This might have to do with how the forces actually work which I'm not familiar with. Please explain to me how this happens. Since I'm a high school student I will be unable to understand the high level math involved(if any in the answer given) so, I would like a conceptual understanding about the situation. </p> | 4,541 | [
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<p>In terms of spectrophotometer, we need to make the light monochromatic before passing it to sample. For that, we use diffraction grating. From the research, I found out that the grating reflects light into different wavelengths. Also, the grating moves in such a way that each wavelength has an opportunity to pass through slit to sample. According to one journal, the monochromatic light is needed to make sure every photon has same energy and wavelength, which means everyone of them, has an equal opportunity to hit the particle. Is there better explanation in this theory? Also, Am I right in thinking that all wavelengths scattered from grating will hit the particles but particles will only absorb certain wavelength?</p> | 4,542 | [
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<p>My textbook says that to find the ket $|ψ\rangle$ in the same position basis as the ket $|ø\rangle$ we do the following: $$|ψ\rangle=\int dø|ø\rangle \langle ø|ψ\rangle$$ Firstly can $|ø\rangle$ be any ket? i.e. this expression just puts $|ψ\rangle$ in the same basis as $|ø\rangle$ regardless of the components of $|ø\rangle$?</p>
<p>Secondly my textbook goes on to say to place $|ψ\rangle$ in the position basis we do the following: $$|ψ\rangle=\int d^3r\ |\mathbf{r}\rangle\langle \mathbf{r}|ψ\rangle$$
Why have we suddenly gained a cubed sign? </p>
<p>Are we taking the integral over nothing? i.e. are the integrals we are doing simply $\int dø$ and $\int d^3r$?</p>
<p>(I am new to this sort of physics/maths and am self teaching so please can you keep the explanations relativity simple) thanks</p> | 4,543 | [
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<p>If <a href="https://en.wikipedia.org/wiki/Carbon%E2%80%93carbon_bond" rel="nofollow">carbon–carbon bonds</a> are reasonably strong and silicon and carbon are both in the same column of the periodic table meaning they have the same amount of valence electrons, also seeing as its bonding with itself electronegativity isn't an issue, why are silicon-silicon bonds weak?</p> | 4,544 | [
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<p>This question is rather historical one.</p>
<p>kilometres can be defined in metres, metres in centimetres, centimetres in millimetres.</p>
<p>There must be some elementary unit (like millimetre or smth.) which cannot be defined in smaller units.</p>
<p>The question is : How does this elementary unit came into being??
e.g. How did scientists decide about exact distance between point A and point B which is considered millimetre??</p> | 4,545 | [
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<p>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.</p>
<p>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?</p>
<p>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.</p> | 4,546 | [
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<p>I tried to understand the importance of conformal transformations in general relativity, but I failed. I didn't see that conformal transformations help to simplify the metrics, and also I didn't see that some physical metric (i.e., metric which describes geometry of some physical system) with conformal scale factor. </p>
<p>Can you give some examples when conformal transformations are physically useful?</p> | 4,547 | [
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<p>When a rock falls from a ledge, why does it head to the surface and not up to where time runs faster?</p>
<p>If a rock, free from forces, follows a worldline of maximum aging, why would that rock approach Earth where the rate of time runs slower, and so would slow down the rocks aging? Shouldn’t the rock avoid earth?</p> | 4,548 | [
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<p>I am looking for a map or plot of the gravitational strength in the solar system. In an ideal world there should be something like google earth to move around in the solar system, zoom in and out and get an visual overview over the field strength in different points in the solar system. </p>
<p>I didn't even find good pictures/plots via google which show different regions of the solar system. So I would appreciate any material about this.</p>
<p><strong>Edit</strong></p>
<p>Here is a map of the solar system which comes close to what I want in the ideal case, but It doesn't have any information about the field strength distribution...</p>
<p><a href="http://www.solarsystemscope.com/" rel="nofollow">http://www.solarsystemscope.com/</a></p> | 4,549 | [
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<p>I have a brief question regarding the formula for wave displacement that I've just encountered. My textbook says:</p>
<p>For a simple plane wave, we have, for a simple harmonic with displacement $u$:</p>
<p>$$u = A \cos(\kappa x - \omega t)$$</p>
<p>where $\omega$ is the angular frequency, $\kappa$ is the wavenumber ($\omega$/wave-velocity), $A$ is the maximum amplitude, and $t$ is time.</p>
<p>Here's my question. If $\kappa$ is defined as $\omega/v$, then I get, by plugging into the formula:</p>
<p>$$u = A \cos(\frac{\omega}{v}x - \omega t)$$</p>
<p>$$= A \cos(\omega (\frac{x}{v} - t))$$</p>
<p>But since $t = \frac{x}{v}$,we get:</p>
<p>$$u = A \cos(\omega(t - t)) = A$$</p>
<p>So by my reasoning (which is obviously wrong), I always end up with $u = A$, which basically makes the whole $\cos$-term redundant. If anyone can explain to me what is wrong with my reasoning, I would greatly appreciate it!</p> | 4,550 | [
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<p>What are the main problems that we need to solve to prove Laplace's determinism correct and overcome the Uncertainty principle?</p> | 49 | [
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