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<p>Let's discuss about $SU(3)$. I understand that the most important representations (relevant to physics) are the defining and the adjoint. In the defining representation of $SU(3)$; namely $\mathbf{3}$, the Gell-Mann matrices are used to represent the generators
$$
\left[T^{A}\right]_{ij} = \dfrac{1}{2}\lambda^{A},
$$
where $T^A$ are the generators and $\lambda^A$ the Gell-Mann matrices. In adjoint representation, on the other hand, an $\mathbf{8}$, the generators are represented by matrices according to
$$
\left[ T_{i} \right]_{jk} = -if_{ijk},
$$
where $f_{ijk}$ are the structure constants.</p>
<p>My question is this, how can one represent the generators in the $\mathbf{10}$ of $SU(3)$, which corresponds to a symmetric tensor with 3 upper or lower indices (or for that matter how to represent the $\mathbf{6}$ with two symmetric indices). What is the general procedure to represent the generators in an arbitrary representation?</p> | g1039 | [
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<p>So in the context of a set of notes I am reading about acoustics I get to equation (23) in this <a href="http://www3.nd.edu/~atassi/Teaching/ame%2060639/Notes/fundamentals_w.pdf" rel="nofollow">paper</a>. Basically it comes down to showing that (<strong>note the dots above the a's meaning time derivative!)</strong></p>
<p>$$
f(t) = -a_0c_0\int_{-\infty}^t \dot{a}(t'+\frac{a_0}{c_0})e^{-\frac{c_0}{a_0}(t-t')} dt'\ = -a_0^2\dot{a}(t),.
$$</p>
<p>under the assumption that $\frac{c_oT}{a_0} >>1$. $T$ can be thought of as a characteristic time scale for this problem. For those interested in more than just the math, please see the paper. Now I show this in the following way:</p>
<p>Integrating by parts:
$$
\begin{align}
f(t) = & \\
=& -a_0^2\dot{a}(t+\frac{a_0}{c_0})+a_0c_0\int_{-\infty}^t \ddot{a}(t'+\frac{a_0}{c_0})\frac{a_0}{c_0}e^{-\frac{c_0}{a_0}(t-t')} dt'\\\
&
\end{align}
$$</p>
<p>By expanding $\dot{a}$ as a taylor series we get:</p>
<p>$$
\dot{a}(t+\frac{a_0}{c_0}) = \dot{a}(t) +\frac{a_0}{c_0}\ddot{a}(t)+O\left(\frac{a_0^2}{c_0^2}\right)
$$</p>
<p>Now if we make the order of magnitude estimate:</p>
<p>$$
\frac{a_0}{c_0}\ddot{a}(t) \simeq \frac{a_0}{c_0T}\dot{a}(t)
$$</p>
<p>then I hope it is clear by applying this reasoning to the equation for $f(t)$ that we have shown $f(t) = -a_0^2\dot{a}(t)$ for $\frac{c_oT}{a_0} >>1$. </p>
<p><strong>Now here comes my question:</strong> How can I justify the time derivative operator behaving as $1/T$, where $T$ was just given as the length scale of the problem (which seems so arbitrary)? Is my reasoning above correct? I was hoping someone could put my mind at ease about the "hand wavy" nature of these order of magnitude approximations. </p> | g1040 | [
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<p>I was wondering with a question for a quite long time, thought to ask here.</p>
<p>I need to know is there any material or element which can block magnetic field? I mean I am searching for such material or element that cannot allow magnetic field though itself?</p>
<p>The practical scenario is, there are two permanent magnets and those are positioned within each other's magnetic field. I want to put something so that both the magnets become free of interference withing themselves.</p>
<p>Hope I could clarify my question.</p>
<p>Can anyone help me of give me some suggestion on this aspect please?</p> | g0 | [
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<p>Why is the gravitation force always attractive? Is there a way to explain this other than the curvature of space time? </p>
<p>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 with concavity?</p> | g81 | [
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<p>I am wondering what will be the physics to explain how two neutral, chemically nonreactive objects stick. I know that using van der Waals formalism, we can treat neutral body electrodynamic forces and arrive with attractive forces that pull the objects together. </p>
<p>Now, once the objects touch (say a mechanical cantiliver in a MEMS sensor like the one used in an iPhone), what happens to the forces? A quantitative answer or some estimate on how strong the attractive force is for simple cases will be very appreciated.</p>
<p>in response to anna's comment :
Let us consider what happens in vacuum for ultra smooth surfaces, with no residual electrical charge and fully chemically stablized surfaces (example, silicon crystals with stabilised surface bonds). </p> | g1041 | [
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<p>I have a question about the relation:
$\exp(-i \vec{\sigma} \cdot \hat{n}\phi/2) = \cos(\phi/2) - i \vec{\sigma} \cdot \hat{n} \sin(\phi/2)$.</p>
<p>In my texts, I see $\phi\hat{n}$ always as c-numbers. My question is whether or not this relation can be generalized for $\hat{n}$ being an operator?</p>
<p>If so how exactly would the expression be different?</p>
<p>Thanks.</p> | g1042 | [
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<p>The second postulate of <a href="http://en.wikipedia.org/wiki/Special_relativity" rel="nofollow">special relativity</a> deals with constancy of light in inertial reference frames. But, how did Einstein came to this conclusion? Did he knew about the <a href="http://en.wikipedia.org/wiki/Michelson%E2%80%93Morley_experiment" rel="nofollow">Michelson-Morley experiment</a>? </p> | g1043 | [
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<p>Does one need to invoke quantum mechanics to explain casimir force or vander waals force. I see that textbooks show derivation of vander waal force with no QM but casimir force is typically described with QM. </p>
<p>Is there a difference between vanderwaal and casimir forces ? Are there distinct examples of these two forces in real life. Is there a way to prove a given force is vanderwaal and not casimir or vice versa. </p> | g1044 | [
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<p>Einstein has suggested that light can behave as a waves as well as like a particle i.e, it has dual character. In 1924, Louis de Broglie suggested that just as light exhibits wave and particle properties, all microscopic material particles such as electrons, protons, atoms, molecules etc. have also dual character. They behave as a particle as well as wave. This means that an electron which has been regarded as a particle also behaves like a wave. Thus, according to de Broglie, </p>
<blockquote>
<p><em><a href="http://en.wikipedia.org/wiki/Matter_wave" rel="nofollow">all the material particles in motion possess wave characteristics</a>.</em> </p>
</blockquote>
<p>Although the dual nature of matter is applicable to all material objects but it is significant for microscopic bodies only. For large bodies, the wavelengths of associated waves are very small and cannot be measured by any of the avalable methods. Therefore, practically these bodies are said to have no wavelengths. Thus, any material body in motion can have wavelength but it is measurable or significant only for microscopic bodies such as electron, proton, atom or molecule. And it is to be noted that electrons won't orbit the nucleus in the sense planet orbits around the sun, but instead exist as <em>STANDING WAVES</em>. </p>
<p>I am finding difficulty in understanding electron or other material particles to behave as wave. In order to understand electron in the form of wave, I considered the following example of high racing car. See the picture below.<br>
<img src="http://i.stack.imgur.com/zcvaI.jpg" alt="enter image description here"></p>
<p>You can see that car is under very less speed compared to electron, it shows some what blur appearance, I considered this to be wave like in nature. And car has comparatively very very less wavelength, so that it can be considered as negligible, even though I considered it to understand. I assumed electron to be under similar position as the car, to understand what actually is called wave nature. Is it that as like car appears to be blur or some what wavy, electron also under immense speed show wave like appearance. But in reality car stays as material particle, even if it appears to be wavy. Is it the same thing what actually meant by wave nature of electron? i.e is it that electron seems to be wavy under high speed, but it will remain as particle during its motion? </p>
<p>This is what I have understood about wave nature of electron. I don't know whether I have misunderstood or not. If I have misunderstood the concept about wave nature, please explain what actually is meant by wave nature of material particles. </p> | g1 | [
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<p>When studying the de Broglie relations, I have stumbled across the following problem:</p>
<blockquote>
<p>Consider an electron with known velocity $v$ and assume: $v \ll c$. Calculate the corresponding wavelength of the electron.</p>
</blockquote>
<p>Using the de Broglie relations from Wikipedia:
$$\begin{align}
\lambda &= h/ p \\
f &= E/h
\end{align}
$$</p>
<p>I would use the first equation and arrive at wavelength $h / (m v)$</p>
<p>However (and this is where my problem is situated) when using the second equation and assuming the only kinetic energy ($ E = m v^2/2$) is present, I arrive at the solution wavelength $2 h / (m v)$, therefore twice as much as in the first case.</p>
<p>My question is: Why do the results differ? I believe it is due to a wrong assumption (kinetic energy not being the total energy) but I can't really prove it.</p>
<p>Thanks in advance!</p> | g2 | [
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<p>Are there any known uses of modeling with elastic fractals in current physical applications? (Especially uses concerning with self-similarity)</p> | g1045 | [
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