title: How to prove the covariant derivative cannot be written as an eigendecomposition of the partial derivative?

text: # The Question How does one prove that Rindler's definition of the covariant derivative of a covariant vector field $\lambda_a$ as \begin{align} \lambda_{a;c} = \lambda_{a,c} - \Gamma^{b}_{\ \ ca} \lambda_{b} \tag{Rindler} \end{align} cannot be written in terms of an eigendecomposition of the partial derivative such as \begin{align} \lambda_{a;c} = \dfrac{\partial y_d}{\partial x_a} \dfrac{\partial }{\partial x_c} \left( \dfrac{\partial x_b}{\partial y_d} \lambda_{b} \right) \tag{linuxfreebird} ? \end{align} # The Definitions The following symbols are defined by Rindler and interpreted by linuxfreebird: \begin{align} \Gamma^{b}_{\ \ ca} = g^{bk} \Gamma_{k ca} \tag{Christoffel symbol} \end{align} are the Christoffel symbols and \begin{align} \tag{inverse metric tensor} g^{bd} = \dfrac{\partial x_b}{\partial y_k} \dfrac{\partial x_d}{\partial y_k} \end{align} is the inverse of the metric tensor $g_{bd}$. The eigendecomposition of a partial derivative is analogous to eigendecomposition of matrix operators. Given a matrix operator $A$, one can express $A$ in terms of its eigenvalue diagonal matrix $\Lambda$ and its eigenvector matrix $V$ such that $A = V\Lambda V^{-1}$. # The Reason for the Question Taken from Rindler, the Riemann tensor $R^{d}_{\ \ abc}$ is defined using the commutator of the covariant derivative from Eq. (Rindler) as \begin{align} \lambda_{a;bc} - \lambda_{a;cb} = R^{d}_{\ \ abc} \lambda_{d} \tag{Riemman tensor}. \end{align} However, if one attempts to calculate the Riemman tensor from Eq. (Riemman tensor) by using the definition of the covariant derivative from Eq. (linuxfreebird), one obtains the following result \begin{align} \lambda_{a;bc} &=& \dfrac{\partial y_n}{\partial x_a} \dfrac{\partial }{\partial x_b} \left( \dfrac{\partial x_m}{\partial y_n} \dfrac{\partial y_f}{\partial x_m} \dfrac{\partial }{\partial x_c} \left( \dfrac{\partial x_d}{\partial y_f} \lambda_{d} \right) \right) \\\&=& \dfrac{\partial y_n}{\partial x_a} \dfrac{\partial^2 }{\partial x_b \partial x_c} \left( \dfrac{\partial x_d}{\partial y_n} \lambda_{d} \right) , \end{align} which states that the commutator of the covariant derivative is zero, hence the Riemman tensor is zero. This was the inconsistency I discovered from Eq. (linuxfreebird) and proves Eq. (linuxfreebird) cannot be an equivalent representation of Eq. (Rindler). However, I have been unable to prove why Eq. (linuxfreebird) is incorrect. Section (The Supporting Material) provides my formulation of Eq. (linuxfreebird). # The Supporting Material Below is my original formulation of Eq. (linuxfreebird): Rindler defines the Christoffel symbol of the first kind as \begin{align} \tag{12} \Gamma_{b ca} = \dfrac{1}{2} \left( \dfrac{\partial g_{bc}}{\partial x_{a}} \+ \dfrac{\partial g_{ab}}{\partial x_{c}} \- \dfrac{\partial g_{ca}}{\partial x_{b}} \right) , \end{align} where \begin{align} \tag{13} g_{ab} = \dfrac{\partial y_k}{\partial x_{a}} \dfrac{\partial y_k}{\partial x_{b}} \end{align} is the metric tensor. The partial derivative of $g_{ab}$ is \begin{align} \tag{14} \dfrac{\partial g_{ab}}{\partial x_{c}} = A_{acb} \+ A_{bca} , \end{align} where \begin{align} \tag{15} A_{acb} = \dfrac{\partial y_k}{\partial x_{a}} \dfrac{\partial^2 y_k}{\partial x_{c}\partial x_{b}} . \end{align} One can express $\Gamma_{bca}$ in terms of $A_{abc}$ as \begin{align} \tag{16} \Gamma_{b ca} = \dfrac{1}{2} \left( A_{bac} + A_{cab} \+ A_{acb} + A_{bca} \- A_{cba} - A_{abc} \right) = A_{bac} , \end{align} which proves that \begin{align} \tag{17} \Gamma_{b ca} = \dfrac{\partial y_k}{\partial x_{b}} \dfrac{\partial^2 y_k}{\partial x_{c}\partial x_{a}} . \end{align} Rindler defines the Christoffel symbol of the second kind as \begin{align} \tag{18} \Gamma^{b}_{\ ca} = g^{bk} \Gamma_{k ca} , \end{align} where \begin{align} \tag{19} g^{ab} = \dfrac{\partial x_{a}}{\partial y_k} \dfrac{\partial x_{b}}{\partial y_k} \end{align} is the inverse of $g_{ab}$. One can express $\Gamma^{b}_{\ ca}$ in terms of the Jacobian components $\partial x_{a}/\partial y_{k}$ and prove the following equivalent expressions \begin{align} \tag{20} \Gamma^{b}_{\ ca} &=& \left( \dfrac{\partial x_{b}}{\partial y_n} \dfrac{\partial x_{k}}{\partial y_n} \right) \left( \dfrac{\partial y_m}{\partial x_{k}} \dfrac{\partial^2 y_m}{\partial x_{c}\partial x_{a}} \right) \\\ &=& \tag{21} \dfrac{\partial x_{b}}{\partial y_n} \left( \dfrac{\partial x_{k}}{\partial y_n} \dfrac{\partial y_m}{\partial x_{k}} \right) \dfrac{\partial^2 y_m}{\partial x_{c}\partial x_{a}} \\\ &=& \tag{22} \dfrac{\partial x_{b}}{\partial y_n} \left( \delta_{mn} \right) \dfrac{\partial^2 y_m}{\partial x_{c}\partial x_{a}} \\\ &=& \tag{23} \dfrac{\partial x_{b}}{\partial y_m} \dfrac{\partial }{\partial x_{c}} \left( \dfrac{\partial y_m}{\partial x_{a}} \right) \\\ &=& \tag{24} \dfrac{\partial }{\partial x_{c}} \left( \dfrac{\partial x_{b}}{\partial y_m} \dfrac{\partial y_m}{\partial x_{a}} \right) \- \dfrac{\partial y_m}{\partial x_{a}} \dfrac{\partial }{\partial x_{c}} \left( \dfrac{\partial x_{b}}{\partial y_m} \right) \\\ &=& \tag{25} \dfrac{\partial }{\partial x_{c}} \left( \delta_{ab} \right) \- \dfrac{\partial y_m}{\partial x_{a}} \dfrac{\partial }{\partial x_{c}} \left( \dfrac{\partial x_{b}}{\partial y_m} \right) \\\ &=& \tag{26} \- \dfrac{\partial y_m}{\partial x_{a}} \dfrac{\partial }{\partial x_{c}} \left( \dfrac{\partial x_{b}}{\partial y_m} \right) . \end{align} Rindler defines the covariant derivative of a covariant vector field $\lambda_{a}$ as \begin{align} \tag{27} \lambda_{a;c} = \lambda_{a,c} - \Gamma^{b}_{\ ca} \lambda_{b} . \end{align} One can define the covariant derivative in terms of $\partial x_{a}/\partial y_{k}$ as \begin{align} \tag{28} \lambda_{a;c} &=& \dfrac{\partial \lambda_{a} }{\partial x_{c}} \+ \dfrac{\partial y_m}{\partial x_{a}} \dfrac{\partial }{\partial x_{c}} \left( \dfrac{\partial x_{b}}{\partial y_m} \right) \lambda_{b} \\\&=& \tag{29} (\delta_{ab}) \dfrac{\partial \lambda_{b} }{\partial x_{c}} \+ \dfrac{\partial y_m}{\partial x_{a}} \dfrac{\partial }{\partial x_{c}} \left( \dfrac{\partial x_{b}}{\partial y_m} \right) \lambda_{b} \\\&=& \tag{30} \left( \dfrac{\partial y_m}{\partial x_{a}} \dfrac{\partial x_{b}}{\partial y_m} \right) \dfrac{\partial \lambda_{b} }{\partial x_{c}} \+ \dfrac{\partial y_m}{\partial x_{a}} \dfrac{\partial }{\partial x_{c}} \left( \dfrac{\partial x_{b}}{\partial y_m} \right) \lambda_{b} \\\&=& \tag{31} \dfrac{\partial y_m}{\partial x_{a}} \left( \dfrac{\partial x_{b}}{\partial y_m} \dfrac{\partial (\lambda_{b} ) }{\partial x_{c}} \+ \dfrac{\partial }{\partial x_{c}} \left( \dfrac{\partial x_{b}}{\partial y_m} \right) \lambda_{b} \right) \\\&=& \tag{32} \dfrac{\partial y_m}{\partial x_{a}} \dfrac{\partial }{\partial x_{c}} \left( \dfrac{\partial x_{b}}{\partial y_m} \lambda_{b} \right) . \end{align}


title: How to integrate over rapidity for Parton Luminosities in LHC?

text: I want to make a comparison of parton luminosities between Tevatron and LHC. According to Factorization theorem the cross section in hadron colliders, as long as the partonic cross section has $\hat \sigma \sim \frac{1}{\hat s}$ dependence, could be written like that : $$\sigma =\sum_{12}\int dx_1 dx_2 f_1 \left(x_1,\mu_F^2\right) f_2\left(x_2,\mu_F^2 \right) \hat{\sigma}_{12} \left(x_1 x_2 s,...\right)$$ here $f_{i}(x_{i},\mu_F^2)$ represents the parton distribution function for $x_i$ fraction of the incoming hadron, $i$ represents the parton flavor and $\mu_F$ is the factorization scale. I wanted to reproduce the Figure 4.4 (page 82) in book Physics at The Terascale or the Figures in LHC Physics Potential vs. Energy. **My Questions?** I appreciate for any reply.   1. When I am scanning over the $\sqrt{\hat s }$ (partonic-center-of-mass energy) what value should I set the factorization scale? $\mu_F = \sqrt{\hat s } $ or $\mu_F = \sqrt{s}$ (center-of-mass energy)?   2. How could I integrate over the rapidity? What I mean is on page 82 it says that "The Luminosity $dL_{ij}/d\hat s$ integrated over y is shown in Figure 4.4a" and I don't understand what changes after integration over y. In Equation 4.11 the parton distribution functions are not function of rapidity thus I could simply move $dy$ to the right side and carry out the integration and it will contribute as a factor (though I don't know the bounds $-\infty<y<\infty$ or something specific for the collider $-5<y<5$ ) however there is relation between the rapidity($y$) and $x$ like this $y=\ln(x1/x2)/2$. Thus I don't understand that line?   3. If I just ignore the rapidity and use the Equation (1) in "LHC Physics Potential vs. Energy" I got some results but I am still off compared to the Figure 1 in the same paper, so I concluded that I don't know how the integration over rapidity affects the over all Parton Luminosity. Equation 4.11 $$\frac{dL_{ij}}{d \hat{s} dy}=\frac{1}{s}\left(x_1,\mu\right) f_2\left(x_2,\mu \right)$$


title: How does the color of fine dispersed metal particles change between silver and black?

text: Small dispersed particles of metals are often black, while a solid object of the same material would be some shade of silver. The most notable example is the use of metallic silver as the black color in classic black and white photography. It is chemically deposited into the gelatine layer on the paper. Another example would be working with metal in a way that creates dust from abrasion (nickel in the example case at hand) - coloring the hands dark gray. So it seems to be a general phenomenon, independent of silver.    So: What is **the cause** of the difference in **appearence depending on particle size**?    I would think it could be a process of dispersion, or possibly destructive interference. Without knowing the details, that would mean the relevant particle sizes should be in the order of the wavelength of visible light? I assume - unverified - that the particles stop being black again with decreasing size at some point. The mechanism causing the silve, reflecting appearence of metal is related to electric conductivity, I expect that could be involved here too.    I will list some questions that occurred to me - they are not explicitly part of the original question: Is an individual particle black, or is the color emergent from the properties of the ensemble?   And at which size does the change happen?   With decreasing size, does the color change from black again? And does it change ti the original silver appearence? Is that dependant on the metal, or particle size only?   Is the transition gradually with size, or abrupt?   Is there anything else that influences the change, so that particles of same size can be silver or black depending on some other condition?   As the black color seems to depend on the interaction of light and opaque particle, it should be fairly independant of the metal - which properties of the metal are relevant?


title: Lagrangian of pendulum with pivot point rotating about a circle

text: I just would like to know If I am on the right track with this stuff. I have scanned the work I have done, sorry if it is a little messy. Disregard the top most $\ddot x$ A disk of radius `R` but with no mass rotates anti clockwise at a constant rate $\omega$. Attached to a point on the edge of the disk is a simple pendulum of length `l` and mass `m`. Use Lagrange's method to find the equations of motion of the mass. ![Page one](http://i.stack.imgur.com/cjc23.jpg) ![Page 2](http://i.stack.imgur.com/u4rnF.jpg)


title: Harmonic oscillator with light damping

text: My textbook gives the following for x as a function of time for a lightly damped harmonic oscillator: $$ x = Ae^{- \gamma t} \cos (\omega \, t)$$ for $\gamma = \dfrac b {2m}$. It says this implies the amplitude follows the relationship: $$x_{max} = Ae^{-\gamma t}$$ I am not understanding where this second equation comes from. Why does the cosine term disappear if $\omega$ is a constant?


title: What is the Earth's magnetic field in space?

text: I have looked around a bit, but haven't found a simple answer. At an altitude between 100-1000 km,   Is the magnetic field B still around $\sim 10^{-5}\ T$? Is there a simple equation that would provide an quick order of magnitude estimation?


title: Higgs Field compared to EM field

text: So, I've been reading about the Higgs because of all of this excitement lately with the LHC. I'm just a layman in physics but one thing I understood was that the Higgs field permeates all of space with a non-zero value. So... Is this like how the vacuum fluctuations don't allow for space to be truly empty? Does the same thing happen to the electromagnetic field so that nowhere in the universe is there zero charge? This QM stuff is deep lol


title: Magnets and speed of light

text: I am in no way a physicist but I do have a fascination with physics. My question is if magnets are being explored / studied as a potential source to achieve the speed of light and if that is even physically possible?


title: Momentum Energy and Higgs

text: So, as an object accelerates it gains energy. And energy is mass. So an object becomes more massive as it approaches the speed of light. But, if mass is ONLY due to an object's interaction with the Higgs field (which I don't fully understand and have another question open about which you're free to answer as well =P ) then how does adding more kinetic energy make it interact more with the Higgs field if you're not adding more particles that interact with the higgs, only energy?????


title: Can two photons annihilate?

text: This is a question about definitions. When two photons interact to create an electron/positron pair, does this process 'count' as annihilation of the photons? I've struggled to find a good definition of the term. Some places say that annihilation requires the end state to be electromagnetic radiation. But, on the other hand, I have found several text books which give annihilation processes ending in hadrons.


title: Conical vs Simple Pendulum

text: I don't understand why the Tension $T$ on a conical pendulum and a simple pendulum are different. In a simple pendulum, one would say that the tension of the rope is $T=mg \cos(\theta)$. ![simple pendulum](http://n8ppq.net/site_n8ppq/Physics/pendulum_files/image001.gif) However, in a conical pendulum (describing a circular motion), $mg=T \cos(\theta)$. ![conical pendulum](http://www.ic.sunysb.edu/Class/phy141md/lib/exe/fetch.php?media=phy141%3alectures%3aconicalpendulum.png) The only difference I see in the set up of the two cases is that in the second one there is a velocity component that makes the bob go around in a circle. I know that in the conical pendulum, the component $T \sin(\theta)$ would give the centripetal acceleration of the circular motion. I've seen this everywhere. The two cases look pretty much the same to me, so I would be tempted to say one of them (rather the second one) is wrong.


title: When can a surface charge density exist?

text: In my syllabus about electromagnetism, they state: "This surface charge density will not always be present, e.g. when considering two non-conducting dielectrics such surface charge density remains absent. However, at a perfect conductor, a surface charge density will be present. If one of the media (or both) carry a conduction current, a surface charge can also be present (explain why!)" I don't really see though, how this can be explained. They seem to imply that there is a relation between the conductivity and the possibility of a surface charge, but I can't figure how they are related. They also state somewhere earlier that there can't be a surface current between two lossy dielectrics. This seems to be analogous to my previous question. I thought that this was due to the fact that dielectrics don't conduct current, but I wonder if someone knows a better explanation here too (that perhaps explains why they explicitly mention that the dielectrics are lossy).


title: What is the significance of electron spin quantum number?

text: Somewhere I read that spin quantum number is a particularly interesting theory of quantum mechanics as what it really implies is that particles like electrons do not come back to the initial state of observation after one rotation, but if they rotate 2 times then the same state is obtained. Is it really so? Even if it/or it is not why do the electrons need two types of spin quantum number? Are two electrons in same orbital supposed to rotate in opposite direction? But then I also hear that electrons are particles and since particles cannot have an axis of rotation they do not actually rotate hence do not "spin". Another proof of this was that the observed magnetic moment of electron is much more than what was expected theoretically from the spinning of electrons hence definitely some phenomenon other than just spinning/rotating must take place.


title: Hamilton's equations for a simple pendulum

text: I don't get how to use Hamilton's equations in mechanics, for example let's take the simple pendulum with $$H=\frac{p^2}{2mR^2}+mgR(1-\cos\theta)$$ Now Hamilton's equations will be: $$\dot p=-mgR\sin\theta$$ $$\dot\theta=\frac{p}{mR^2}$$ I know one of the points of Hamiltonian formalism is to get first order diff. equations instead of second order that Lagrangian formalism gives you, but how can I proceed from here without just derivating again wrt. $\dot\theta$ and substituting $\dot p$ to get the same equation that I get with the Lagrangian formulation? Or is that the way to do it? And how could I get the path of the system on the phase space with those equations?


title: Easy to understand equivalents for energy (kilowatt-hour)

text: Is there a database/webpage/other source for obtaining simple, easy to understand and meaningful equivalents for converted energy by a photovoltaic- site (Wh/kWh/MWh/GWh)? Something like "xxxkWh equals watching the superbowl on a lcd screen for 1 hour"? Just simple stuff, not too technical - for the non technical mere mortals so to speak.


title: Correct way to do a Thomas-Fermi approximation for cold gases

text: I have calculated the total Gross-Pitaevskii energy for a 2D Bose-Einstein condensate in an harmonical trap, using a variational gaussian wave function with a variational parameter _b_. Now I want to compare the variational energy to the Thomas-Fermi result. I know that the Thomas-Fermi approximation means that you neglect the total kinetic energy in comparison to the interaction energy, but I was wondering how to do it specifically in this case. I namely have three different possibilities in mind: 1) Just remove the kinetic energy term from the energy expression I found with the variational wave function, and keep the value of the variational parameter _b_ as it was before. 2) Remove the kinetic energy term from the energy expression I found with the variational wave function, and calculate a new value for the variational parameter _b_ for this specific case. 3) Use the Thomas-Fermi approximation in the GP equation to find a new expression for the wave function (instead of the one I used before) and use this one to calculate the energy. I can't seem to decide which of these three is the right one. Can anyone give a convincing argument as to which method I should use?


title: What are independent parameters in Hellmann–Feynman theorem?

text: A typical example in textbooks about the application of Hellmann–Feynman theorem is calculating $\left\langle\frac{1}{r^2}\right\rangle$ in hydrogen- like atoms. Wikipedia has a nice demonstration of this. At some point in the Wikipedia derivation is used that   $$\frac{\partial n}{\partial \ell}~=~1. \qquad\qquad (1)$$ But why is eq. (1) true? I know that $$n=n_r+\ell+1,$$ but $n_r$ is just another variable with different physical meaning, so why is $n_r$ independent from $\ell$, whereas $n$ is not? The Wikipedia proof for Hellmann–Feynman theorem does not address the problem of independence of different parameters. What variables are kept fixed during the differentiation $(1)$ and why? The Wikipedia page seems to have only a vague notion of $\frac{\partial\hat{H}}{\partial\lambda}$ and $\frac{\partial E}{\partial\lambda}$, unlike in, e.g., thermodynamics, where all partial derivatives are typically written like $$\left( \frac{\partial U}{\partial V} \right)_S \qquad ,\qquad \left( \frac{\partial U}{\partial V} \right)_p \qquad ,\qquad \ldots $$ so that it is clear which variables are kept fixed during the differentiations.


title: Can x-ray radiation be compared to background radiation?

text: I've been trying to learn about the possible effects of x-ray radiation from dental x-rays and most of the resources I come across compare the exposure to that of natural background radiation. Here's an example (not specific to dental x-rays): http://www.hpa.org.uk/webw/HPAweb&HPAwebStandard/HPAweb_C/1195733826941?p=1158934607708 I know electro-magnetic-radiation comes in different wavelengths and some are ionizing and some are not. Is this a valid comparison? **Update:** Thanks for the helpful responses so far. Looking at the chart here from @anna v, is comparing x-rays with background radiation comparing EMR that is 10^-10 with wavelengths that are longer than infrared? That's what I still don't understand. Wouldn't those different types of waves have different properties?


title: Most natural tensor structure for a quantum field

text: A quantum field is described by a Hilbert space. In many instances, the chosen tensor structure on this Hilbert space corresponds to that of space-like separated regions of space-time. The justification to this is that we can only perform local measurements, so the tensor structure should correspond to what subsystems operationally are, e.g. space-like separated regions of space-time. Does any one know a paper where the choice of a tensor structure for quantum fields is better motivated, and alternative tensor product structure proposed (for example in the context of relativistic quantum information, where space- like separated regions should not be the fundamental unit anymore since communication is generally allowed)?


title: Boltzmann entropy and phase space volume

text: In Huang's book on Statistical mechanism and Statistical interpretation of Entropy, it is not mentioned that $\Omega$ is the phase space volume, but it is the states of the system. So, how does entropy be called as logarithm of phase space volume? A detailed explanation and reference where it is shown that Entropy as phase space volume will be appreciated. Where can I find the derivation that entropy is the phase space volume?


title: Diagonal stress energy tensor components

text: If I got a diagonal stress energy tensor $T_{\alpha \alpha} = x_{\alpha}$ for some coefficients $x_{\alpha}$, could anyone tell me how can I extract the four components of the stress energy tensor. Also, is there a way to derive the metric for such a case. Thanks in advance.


title: Grand canonical ensemble with interaction, simulation doubts

text: First of all, this is a follow-up of my first question. The idea is the same, every state would consist of some particles in a line (with an energy associated) and the particles can not be nearer than $D$ units between each other. Now I have to add the possibility that the number of particles changes (it is of course not greater than $\frac{M}{D}$) and I thought of using the grand canonical ensemble, but I have yet again some doubts. First, why is it not more likely for such an ensemble to have 0 particles than more than 0? I mean, I see in the formula that $P \propto e^{\frac{\mu N - E(s)}{kT}}$ but I don't understand why, if it costs energy to get a new particle, the probability is larger. What am I failing to see? Second, how can $\mu$ be derived, how would you derive it in a normal experiment? This is not totally necessary (I can simply adjust it until my simulations match the real data) but I would like to know as an illiterate. Finally, for the Boltzmann distribution case, I used a Metropolis-Hastings algorithm to approximate the distribution function. Is it advisable here as well or is there any other better method? Thank you very much for your help!


title: How "unnatural" is the universe?

text: Looking to read up on the impact the discovery of Higgs boson has on the String Theory I came upon these two paragraphs in an article about the Higgs boson Nobel Prize: > One possibility has been brought up that even physicists don’t like to think > about. Maybe the universe is even stranger than they think. Like, so strange > that even post-Standard Model models can’t account for it. Some physicists > are starting to question whether or not our universe is natural. This cuts > to the heart of why our reality has the features that it does: that is, full > of quarks and electricity and a particular speed of light. > > This problem, the naturalness or unnaturalness of our universe, can be > likened to a weird thought experiment. Suppose you walk into a room and find > a pencil balanced perfectly vertical on its sharp tip. That would be a > fairly unnatural state for the pencil to be in because any small deviation > would have caused it to fall down. This is how physicists have found the > universe: a bunch of rather well-tuned fundamental constants have been > discovered that produce the reality that we see. I thought this was a gross exaggeration of how weird and unnatural the universe is (it was after all written by someone who starts his sentences with "Like") so I wanted to get other opinions on whether the state of our universe is really as weird as "finding a pencil balanced perfectly vertical on its sharp tip"?


title: Covariant derivative with upper index

text: I just need clarification, that is, to see that I'm doing the right thing. When calculating central charge for certain metric, I need to solve an integral that contains Lie brackets etc. And I have some expression, that contains terms like: $$\xi_\sigma D^\nu h^{\mu\sigma}$$ Where $D_\nu$ should be covariant derivative. Now, can I just say: $$D^\nu=g^{\nu\mu}D_\mu$$ And similarly for the $h^{\mu\sigma}$, and use the standard definition of covariant derivatives?


title: When motion begins, do objects go through an infinite number of position derivatives?

text: This might be a very vague and unclear question, but let me explain. When an object at rest moves, or moves from point $A$ to point $B$, we know the object must have had some velocity (1st derivative of position) during that trip. It's also true that the object had to have accelerated to gain that velocity (2nd derivative)... if we extend this argument, is it logical to say that the object must have experienced jerk (3rd deriv.) to gain that acceleration? And so on and so forth... 3rd, 4th, 5th position derivatives? Or is my logic flawed somewhere?


title: Concept map/graph

text: Where can I find maps of physical concepts illustrating the connection & logical relations between concepts . I think it can be of a great help to summarize what I learnt .


title: Possible Deflection Distance For Falling Object

text: I work for a company selling tethers that prevent tools on building sites from falling away from the user. One of the problems that has come up is that objects can deflect horizontally when they strike an object whilst falling. You can imagine that they've fallen and then hit a steel girder or something on the way down. Given that the weight of the object, the height dropped from and the impact height are known, how would I calculate the _possible_ deflection distance?


title: Lie Algebra Generators

text: We know that rotations are performed via real and orthogonal matrices, $O^{T}=O^{-1}$. We can write $O$ as, (The proper rotations have unit determinant) $$O = \exp(A),$$ where $A^{T}=-A$. In three space dimensions a real anti-symmetric matrix will require three independent entries. $$A_{ab} = \epsilon_{abc}\theta_c.$$ This looks like $\vec{M}\cdot\vec{\theta}$, where the matrix elements of the generators $M_c$ are given as $$(M_c)_{ab}=\epsilon_{abc}.$$ Let us remember: $$[t_a,t_b]= i\,\,f_{abc}t_c.$$ Where $f$ is structure constant. We know that $f$'s should satisfy _Jacobi identity_ $$f_{bcm}f_{mah} + f_{abh}f_{hcm} + f_{cah}f_{hbm} = 0.$$ We also know that: $$t_a\,t_b - t_b\,t_c - i\,f_{abc}t_c = 0.$$ The question is: How can I find $t_a$ individually in terms of $f$'s? Do you have a suggestion?


title: Lie algebra - basis for adjoint matrix products in SU(N)?

text: In $SU(N)$, the set of matrices in the fundamental representation plus the identity: $$ \left\\{ \mathbf{1}, t^a \right\\} $$ acts as a basis for generator products under matrix multiplication, such that any combination of the elements in the set can be written as a linear combination, i.e. for any $$ X_F = t^{a_1}\dotsm t^{a_n} $$ we can write $$ X_F = c^0 \mathbf{1} + c^a t^a \, .$$ It is easy to show that $$ c^0 = \frac{1}{N} \text{tr} (X_F) \, , \quad c^a = - 2 i\, \text{tr} (X_F t^a)\, . $$ We cannot do the same for the adjoint representation because the set $$ \left\\{ \mathbf{1}, T^a \right\\} $$ doesn't span the full product space, e.g. $T^a T^b$ cannot be expressed solely in terms of $\mathbf{1}$ and $T^c$. So my question is: can we add - and if yes which - a minimal number of $n^2\\!-\\!1$ matrices $\\{K^a\\}$ to $ \left\\{ \mathbf{1}, T^a \right\\} $ in order to make it a closed set? **EDIT** Where it comes from is that because in the fundamental representation we have additional anti commutation rules $$ \left\\{ t^a, t^b \right\\} = \delta^{ab} \frac{\mathbf{1}}{N} + d^{abc} t^c \, , $$ we can take the mean of these and the standard commutation rules to write $$ t^a t^b = \frac{1}{2} \delta^{ab} \frac{\mathbf{1}}{N} +\frac{1}{2} h^{abc} t^c \,,$$ where $h^{abc}=d^{abc}+i f^{abc}$. We chain this relation to write for any product of fundamental generators $$ t^{a_1}\cdots t^{a_n} = A^{a_1\cdots a_n} \frac{\mathbf{1}}{N} + B^{a_1\cdots a_n b} t^b \, ,$$ with $A$ and $B$ some constants. This is the same statement as above. However we cannot do the same in the adjoint representation, because then the anti commutation rules don't close, i.e. they cannot be written as a linear combination of $\mathbf{1}$ and $T^a$. So the question is, can we add some extra matrices $\\{K^a\\}$ such that we can write the anti commutation rules as $$ \left\\{ T^a, T^b \right\\} = \delta^{ab} \frac{\mathbf{1}}{N} + d^{abc} T^c + c^{abc}K^c\, , $$ which would give $$ T^{a_1}\cdots T^{a_n} = A^{a_1\cdots a_n} \frac{\mathbf{1}}{N} + B^{a_1\cdots a_n b} T^b + C^{a_1\cdots a_n b} K^c\, .$$


title: How can Lorentz contraction be "directly" measured?

text: If one is watching a relativistic object of e.g. spherical shape, which emits enough light to be detectable, it will, despite being Lorentz contracted, appear of its natural shape, although rotated. This phenomenon is called Terrel rotation$^\dagger$. Citing wikipedia on Lorentz contraction, "length contraction is the phenomenon of a decrease in length measured by the observer of an object which is traveling at any non-zero velocity relative to the observer". So, how can the observer _actually measure_ this decrease in length? Can it be somehow done in a non-relativistic regime of a measurement apparatus? $^\dagger$Russian version of the page gives more detail with some pictures


title: Bohmian loophole in PBR-like theorems

text: I'm reviving and expanding this question, because of the new paper today, by Aaronson et al. The more general question is: **How does quantum-potential Bohmian mechanics relate to no-go theorems for psi-epistemic theories?** According to Bohmian mechanics, there is a wavefunction as in conventional quantum mechanics, which evolves according to a Schrödinger equation, and then a set of classical degrees of freedom, which evolve according to the gradient of the phase of this wavefunction. A Born-rule probability distribution for the classical degrees of freedom is preserved by this law of motion, so Bohmian mechanics provides a deterministic hidden-variables theory, albeit a nonlocal one. However, the equations of motion for the classical variables can be rewritten in terms of two potentials, the local potential appearing in the Schrödinger equation, and a nonlocal "quantum potential" which depends on the form of the wavefunction. This means that Bohmian mechanics _for any given wavefunction_ can be replaced by a nonlocal deterministic theory in which there is no wavefunction. (Note that the quantum potential will be time-dependent unless the wavefunction was an energy eigenstate; thanks to Tatiana Seletskaia for emphasizing this point.) Meanwhile, there has recently been theoretical interest in ruling out so- called "psi-epistemic" hidden-variables theories, in which the quantum wavefunction is not part of the ontology. Obviously, "quantum-potential Bohmian mechanics" is a psi-epistemic theory. It ought to be relevant to this enterprise. But it may require some mental gymnastics to bring it into contact with the PBR-like theorems, because the mapping from quantum mechanics to these quantum-potential theories is one-to-many: scenarios with the same quantum equations of motion, but different initial conditions for the wavefunction, correspond to _different equations of motion_ in a quantum- potential theory. NOTE 1: Empirically we only have one world to account for, so a real-life quantum-potential theorist in principle shouldn't even need their formalism to match the whole of some quantum theory. Quantum cosmology only needs one wavefunction of the universe, and so such a theorist would only need to consider one "cosmic quantum potential" in order to define their theory. NOTE 2: Here is the original version of this question, that I asked in January: The "PBR theorem" (Pusey-Barrett-Rudolph) purports to show that you can't reproduce the predictions of quantum mechanics without supposing that wavefunctions are real. But it always seemed obvious to me that this was wrong, because you can rewrite Bohmian mechanics so that there's no "pilot wave" - just rewrite the equation of motion for the Bohmian particles, so that the influence coming from the pilot wave is reproduced by a nonlocal potential. Can someone explain how it is that PBR's deduction overlooks this possibility?


title: 2nd - 3rd Year Physics Labs

text: I'm looking to propose a new physics laboratory experiment, does anyone have any particularly fond memories of an experiment they carried out whilst at University? I'd like to include a section that makes use of Mathematica, Arduino or Python. The current labs are lacking computational aspects. The lab would ideally run over three 4 hour sessions.


title: Equation of motion in quantum system

text: The commutation relation between H and $L_i$ is $[H,L_i]$. I want to show the equation of motion. **Is this right: $ [H,L_i] = -i \frac{dL_i}{dt}?$**


title: Stability of a rotating ring of multiple electrons at relativistic speeds

text: There was a time when physicists where concerned about electron internal structure. The rotating ring model was one of the proposals to explain how a charge density could become stable against electrostatic repulsion, by postulating that the charge is distributed as a ring with angular velocity, which engenders a magnetostatic force that exactly cancels the electrostatic repulsion (actually they don't exactly cancel, they both become equal to the centripetal force) Now we don't concern ourselves with the electron internal structure anymore, but i wonder if there are other physical circumstances where we can still see a ring of charges to become stable against self-repulsion by this mechanism One of the problems i see is that it is usually assumed in the analysis that the ring rotates at the speed of light. My question is: > Is possible to stabilize such structure by physical sub-light tangential > velocities of the ring? **Edit** There have been some research on a related area of nuclear phenomena, namely high spin nuclear deformation (a book reference about such excitations here), a very interesting application of the above ideas is to look out to reduce or eliminate nuclear decay processes in unstable nuclei, by applying a high spin with a strong background magnetic field to such nuclei, something i highlighted in the comments to this answer.


title: Should a pen be light or heavy?

text: Expensive pens are usually heavy. Naturally a heavy weight is not expected for an ordinary pen. But in the view of mechanics, the more the inertia the more the capability of holding the speed. Is then the reason for making pens heavy?


title: How does the moon reflect light?

text: We can see the moon in the night because it reflects sunlight. But the light is incident on the opposite side of moon with respect to the observer in the night. In this case, how does the moon reflect light? I am not sure if I have articulated my question correctly.


title: Why must these Spinors be normalized?

text: I have just begun studying spin and there are two spinors mentioned: The main spinor $\chi $ and the spin-up spin down spinors (eigenspinors) $\chi_+ ,\chi_- $. I learned that the main spinor is a linear combination of the two spin spinors and I understand that the main spinor must be normalized. I don't understand however why the spin-up and spin-down spinors must also be normalized. As far as I understand, they are not involved in probability calculations for example: $\chi_+^{(x)} = \left( \begin{array}{ccc} \frac{1}{\sqrt2} \\\ \frac{1}{\sqrt2} \\\ \end{array} \right)$ instead of just $\chi_+^{(x)} = \left( \begin{array}{ccc} 1 \\\ 1 \\\ \end{array} \right)$ What am I missing?


title: Perfect Ideal Gas Modelization of Air

text: I'm solving a Thermodynamic problem. I have Air at temperature of 293.15 K, and pressure of 20 MPa. I would understand if is possible to consider this an Ideal Gas or not (and if is possible which are the limit of pressure and temperature with i can still consider air as an Ideal Gas). Thankyou very much. Samuele


title: Phase and amplitude information of an image

text: By applying Fourier Transform to an image we can get its magnitude as well as phase spectrum. A magnitude spectrum describes how various frequencies are attenuated and accentuated in that image but what information does the phase spectrum actually gives? 2D images records the amplitude information alone unlike 3D images which gives both the amplitude as well as phase information. Then what is actually this phase information we get by applying Fourier Transform to an image?


title: Space-time curvature creates gravity or is it (could it be) vice-versa too?

text: Mass (Energy) creates space-time curvature and thus it forms the reason for gravity. Can it be vice-versa too? Like, mass created gravitational field, gravitational field created space-time curvature? I would like to agree, because, in that sense, I can say electromagnetic field also produces space-time curvature! This can lead to a unification between them! (that's worth a question)


title: Walking & Swinging

text: How can I show that the most convenient way to move the arms while walking is swinging them back and forth, alternatively? To pose the question in another way: can I prove, starting from the conservation of momentum and angular momentum, that given a rigid solid body moving at constant speed and with two appendices on the side, this appendices will move as our arms do when we walk "freely"? ![enter image description here](http://i.stack.imgur.com/O7b0z.jpg)


title: Would there be time dilation at the point where two gravitational fields cancel each other out?

text: My question is very simple, and most likely a stupid one: One observer is at a point in space were the gravitational force form massive bodies (or a single massive body) cancel each-other out. The second observer is in another hypothetical situation where there exists no massive bodies and therefore no gravity. My question is this, is there a relativistic time difference between the two?


title: Is cosmic background radiation dark-matter and/or dark-energy?

text: Dumb question alert: Is it possible that the cosmic background radiation might be the source of dark-matter and/or dark-energy? What is the mass of the background radiation in the known universe?


title: What happens to string theory if spacetime is doomed?

text: What is expected to happen with string theory, if physics is reformulated according the lines hinted at by the twistor-uprising business discussed in this question and its answers for example and spacetime is doomed, as Nima likes to say? Will it be incorporated into this new picture with "emergent" spacetime and surface again in certain limits? Or more generally, what effects will this twistor-revolution have on string theory, if any?


title: Defining a local polarization field in a distribution of charge

text: I am currently building a theoretical model where charges of opposite signs are created by pairs and then diffuse and are drifted by an electrical field. I am taking this along a single so far, for simplicity. The object I'd like to arrive at the end is a local (density of) polarization, and not the global one (because the diffusion gives me two gaussians, spreading in time and drifting in opposites ways. But when I try to come up we've an expression for this polarization field, nothing pertinent comes out: if I look at one point and compute the barycenter of the charges from there, I get the same expression no manner where it is because of the global charge neutrality; and when I think about only computing charges in a small area about the point, I can't guarantee the neutrality of the charges inside this area (and so polarization is irrelevant in there). I am wondering if it's not because there is not more information than the one given by the separation of the two charge distribution barycenters.


title: How do rudders work?

text: It is possible for a boat to use rudder to make a U-turn while coasting (moving by inertia), although it would lose some speed. How exactly do boats trade a portion of magnitude of the initial linear momentum into a direction change? I would expect that the only force there is friction which would only reduce magnitude of the momentum, but it is actually changes its direction. It must have to do with rudders, but I must admit I don't understand how they work.


title: Perturbation theory emitting high order powers

text: For my second-order energy correction for a harmonic oscillator in an electric field I have the following: $$q^2\varepsilon^2\sum_{m\neq n}\frac{|\langle m|x|n\rangle|^2}{E^{(0)}_n-E^{(0)}_m}+\text{ }...$$   1. My first question is what values of $m$ do we sum over?    2. Secondly, my textbook says we neglect higher-power terms, does this mean higher values of $m$ or what?  If I have left any information out (which is very likely) just say.


title: Why are atomic quadrupole moments calculated using nuclear spin?

text: It's my understanding that electric quadrupoles interact with the gradient of an electric field, and I understand roughly how this works. I am trying to calculate the interaction between an atomic nucleus and a field gradient using this understanding. However, this does not seem trivial and from what I've read (in textbooks and online) **an atom's nuclear quadrupole moment is directly proportional to its total spin**. Why is this? Sources: According to Wikipedia, nuclei have electric quadrupole moments if their total spin is 1 or greater. Here is one source for the nuclear quadrupole moment formula that is fairly easy to read. (Most of my sources so far are textbooks, which are not public-accessible, but the above source explains things fairly well.) **Edit** : As far as I can tell, the formula for the energy of a quadrupole in a field gradient is very close to this one (but potentially different, which is why I'm trying to derive it): $E_Q = \frac{e V_{zz} Q} {(4 I) (2 I - 1) {\hbar}} [(3 I_z^2 - I^2) + {\eta} * (I_x^2 - I_y^2)]$ where:   * $Q$ is the atom's quadrupole moment   * $I_x$ its x spin (etc)   * $V_{zz}$ the z field gradient   * ${\eta} = \frac{V_{xx}-V_{yy}}{V_{zz}}$ the "asymmetry parameter". As you can see, it is heavily dependent on spin.


title: How do I calculate integer and half integer spin?

text: How do I calculate integer and half integer spin, and how do I use the calculations?


title: Why isn't invariant notation common?

text: In principle, one can write quantities in a manifestly invariant - rather than covariant - fashion in e.g. special relativity. For example, rather than writing just $x^\mu$, we could write the basis explicitly, and ask that the basis transforms oppositely to the components, $$ x = x^\mu e_\mu^{(i)}, $$ such that $x$ is invariant. Why isn't such an invariant notation more common in e.g. special and general relativity? Do mathematicians also work with covariant language? or the invariant language? Is it because the basis disappears in contractions if the basis is orthonormal? e.g. $$ x(y) = x^\mu e_\mu y_\nu e^\nu = x^\mu y_\nu \delta^\nu_\mu = x^\mu y_\mu $$


title: Twinkling in sunlight reflections

text: A question that covers optics and possibly biology/physiology: looking out my window, I see the Sun reflecting off car windshields in a distant parking lot. The very bright reflections are distinctly twinkling. Is this twinkling false, and it's actually irritation to my eyes from the brightness (as well as tiny eye movements [saccades])? Or is it really twinkling, and if so, why? Other objects at that distance seem pretty steady. If the atmosphere is churning enough to cause the solar reflection to twinkle (either before or after the reflection), I would think everything else would twinkle too. Could the Sun heat up the cars enough to cause extra turbulence close to them, causing twinkling just in the line of sight to the reflection? If I had a pair of binoculars, I could consider checking out whether the rest of the car is twinkling (but would have to be careful about eye injury from the Sun's reflection!).


title: Need help simulating a fan

text: I'm trying to write some code to approximately simulate a number fans in an arbitrarily shaped room. I'm not real strong in physics, and could use some help. If it helps I'm not interested in fine grain simulation and I'm only interested in simulating on 2 dimensions instead of 3. What I have so far is an array that can hold information about the air (pressure, whatever else I need to) I'm having trouble thinking through what exactly a fan does. My thoughts so far is that a fan takes air from one side and adds it to the other side. This creates a lower pressure on one side and a higher pressure on the other side, but it seems like a fan does more than just that, because there is a stream of air on the output side that doesn't exist on the input side. If it were just high pressure and low pressure, it wouldn't stream but just diffuse (if that's the right word). So my gut says there is something else going on, but I don't know what it could be or what I need to model it. From what I've gathered on wikipedia this falls under fluid dynamics even though I'm not asking about fluids, so I've tagged it with that.


title: Why don't we consider electrostatic energy of the pair in the case of pair production?

text: I have seen this Wikipedia article and many others, but in none of them I find any mention of the **electro-static energy** of the generated pair. Why? I mean, the energy conservation should be written as $h\nu = E_+ + E_- + Electrostatic\ Energy$


title: Looking for lists of student difficulties with electromagnetism, circuits (DC and AC)

text: I'm trying to help with an Instructor's Guide for an introductory calc-based physics text, and it would really help if I could locate some lists of typical student difficulties with topics like ... flux induced E and B fields Faraday's law Lenz's law DC circuits AC circuits Is there a better approach than just Googling "student difficulties with XXX," as I have been trying with rather inconsistent luck? _Added--_ Thanks for the reminder! I actually found a copy of Randy Knight's book in the attic that I got from him at an AAPT meeting some years back. I can't believe I actually located it in my chaos ... great suggestion!


title: Quantum Mutual Information scaling

text: Wikipedia provides a simple definition of Quantum Mutual Information: $$I(\rho^{ab})= S(\rho^{a}) + S(\rho^{b}) - S(\rho^{ab})$$ where in terms of relative information we have: $$I(\rho^{ab})= S(\rho^{ab}|| \rho^{a} \otimes \rho^{b})$$ what is the best notational approach for these expression when there are more than two systems? UPDATE ![Information Relationships](http://i.stack.imgur.com/FKesZ.jpg) If we take the classical case: $$I(X;Y) = H(X \cap Y)$$ The form of Pascal's triangle is apparent when the relation is viewed as (1 2 1): $$0=- H(X\cup Y) + \\{ H(X) + H(Y) \\}- H(X\cap Y)$$ and (1,3,3,1): $$0=- H(X\cup Y\cup Z) + \\{ H(X) + H(Y) + H(Z)\\}-\\{ H(X\cap Y) + H(X \cap Z) + H(Y \cap Z)\\}+ H(X\cap Y\cap Z)$$ Following the inclusion-exclusion principle we find that the number of terms that comprise mutual information are: $$\sum_{k=2}^{m}(-1)^{k-1}\dbinom{m}{k}$$ Where we could write the complete expression as: $$H\left(\bigcup_{i=1}^{n}X_i\right)=\sum_{i=1}^{n}H(X_i) - \sum_{i,j:i<j}H(X_i\cap X_j) + \sum_{i,j,k:i<j<k}H(X_i\cap X_j \cap X_k) -\dots + (-1)^{n-1}H\left(\bigcap_{i=1}^{n}X_i\right)$$ or as: $$H\left(\bigcup_{i=1}^{n}X_i\right)=\sum_{i=1}^{n}H(X_i) + \sum_{k=2}^{n}(-1)^{k-1}\sum_{I\subset \\{1,\dots,n\\}|I|=k} H\left(\bigcap_{i\in I}^{n}X_i\right)$$ which highlights that the mutual information is associated with terms: $$I\left(\bigcap_{i\in I}^{n}X_i\right)= \sum_{k=2}^{n}(-1)^{k-1}\sum_{I\subset \\{2,\dots,n\\}|I|=k} H\left(\bigcap_{i\in I}^{n}X_i\right)$$ If we follow the same form then the apparent form for quantum mutual information would would also follow pascal's triangle's binomial structure, so what is the most efficient notation to use?


title: Storing light energy as potential kinetic to drive an electric generator

text: I would like to lift a 10 ton mass (crate of rocks) to a height of about 80 feet using a small 24 volt electric motor powered by a 16 square foot photovoltaic panel. Then I would like to collect the energy stored by this 10 ton mass by having it turn an electric generator in a controlled descent. One of the problems I must solve it finding a gear assembly that will allow this small 24 volt motor to raise such a large mass. I would like to get information related to mechanics of such gear assembly and it would be just great if someone could direct me to a company who builds such gear assemblies.


title: How to Make RF Waves Visible

text: I understand RF (Radio Frequency) Waves are electromagnetic waves and a mode of communication for wireless technologies, such as cordless phones, radar, ham radio, GPS, and television broadcasts. Most modes of RF waves are invisible. Can RF Waves be visible? If so, how do I make RF Waves visible? Is an X-ray an example. Please give me a few examples?


title: No monopoles in the Weinberg-Salam model

text: I'm reading Chapter 10.4 on the 't Hooft-Polyakov monopoles in Ryder's Quantum Field Theory. On page 412 he explains why magnetic monopoles cannot appear in the Weinberg- Salam model. I'm I right by saying that he shows that the electromagnetic gauge group $U(1)_{em}$ is not compactly embedded into the $U(1)\times U(1)$ subgroup of $SU(2)\times U(1)$? He then immediately concludes that the first fundamental group of the unbroken symmetry, which is $H=U(1)_{em}$, $\pi_1(H)$ must be trivial or doesn't exists. Could someone refer me why? Comment: I know that in the $SU(2)\times U(1)$ ones must consider the second homotopy group from $S^2$ to the orbit $G/H=SU(2)\times U(1)/U(1)$, where $H$ is the isotropy group of a vacuum state, after symmetry breaking. But the second homotopy group of a quotient can be related through a exact series to the kernel of the map from $\pi_1(H)$ into $\pi_1(G)$. What I do not understand is by which theorem for $H$ having a non-compact covering group $\pi_1(H)$ must be trivial or non-existing (???)?


title: What is the difference between $|0\rangle $ and $0$?

text: What is the difference between $|0\rangle $ and $0$ in the context of $$a_- |0\rangle =0~?$$


title: Gravity fed containers

text: Curiosity got the better of me, and was wondering if you have two containers full of sand each at different heights but both fed in a tube from a with the containers being in Y configuration, is there any difference in which one drains first? ![Sand](http://i.stack.imgur.com/LA2bW.gif)


title: Induction and Halbach Array

text: I have recently studied and tried to understand the concept of the Inductrack ( http://en.wikipedia.org/wiki/Inductrack ), a form a magnetic levitation. In the following link they present three different ways to construct the track: http://cegt201.bradley.edu/projects/proj2004/maglevt1/inductrack.html The first one is pretty straight forward, basically utilizing wire coils as you would expect from any simple induction experiment. But with the other two methods I have a bit of trouble understanding the workings behind them, hopefully someone here can help. The third method (utilizing the laminated slotted copper sheets):   from what I understand this would basically be the same as horizontally aligned wire coils and due to the sinusoidal nature of the magnetic field there is still induction when the array is moved.   Is this correct?   If so, is the orientation of the wire coils completely irrelevant to the effectiveness of the system?   Or is there an alignment that would be optimal? This brings me to the second method utilizing wire rungs.   Here every wire is shortened to every other wire in the pack. Is this setup better than any of the other two in terms of effectiveness? Which would physically speaking be the ideal way to construct the track? Thank you very much in advance for your help, time and effort, it is much appreciated! Have a great day, Tom


title: Integers powers of fields in a QFT Lagrangian

text: Why can we not have non-integer powers of fields in a QFT Lagrangian, eg. $\phi^{5/2}$? Or if we wanted a charged $\phi^3$ theory, could we not have a $(\phi^*\phi)^{3/2}$ term?


title: Vertical Motion in Air + Density

text: I am having trouble with this confusing question. Please help. The fastest recorded skydive was by an Air Force officer who jumped from a helium balloon at an elevation of 103000 ft, three times higher than airliners fly. Because the density of air is so low at these altitudes, he reached a speed of 614 mph at an elevation of 90000 ft, then gradually slowed as the air became more dense. Assume that he fell in the spread-eagle position and that his low-altitude terminal speed is 125 mph. Use this information to determine the density of air at 90000 ft. Thanks.


title: Temperature vs AC energy consumption

text: I'm need to understand the following: to keep the room at confortable temperature (70 degree, for example), how does the amount of energy consumed by the AC grow as the outside air temperature rises in the summer over the same length of time (a day, say)? Is it more or less linear? quadratic? exponential? or anything else? I posted the same question on another StackExchange site, but I figured that physicist may provide more help because it seems to me that it is related to both heat conduction and the working mechanism of AC. Thanks.


title: why isn't the variance of Fano noise proportional to the number of photon arrivals?

text: I've been reading "Photon Transfer" by James Janesick and in Chapter 3 (http://spie.org/samples/PM170.pdf) he describes the various sources of noise in a CCD. Basically, incoming photons interact with the CCD and cause some electrons to jump the bandgap. Both of these processes have some inherent randomness and must be described statistically - the photon arrival rate's variability gives rise to the shot noise, while the variability in electrons produced from a given photon impact gives rise to the Fano noise. The photon arrival is a Poisson process, so if m is the expected number of photon arrivals during a given exposure, the actual number of arrivals is a Poisson distributed random variable N(m), and var(N) = m. Now here's the part I don't understand. I quote from Janesick, editing slightly for clarity (p. 26): > If all the energy of an interacting photon was spent in the production of > electron-hole (e-h) pairs, then there would be no variation in the number of > e-h pairs produced. On the other hand, if the energy was partitioned between > breaking covalent bonds and lattice vibrations, or if phonon production was > uncorrelated, Poisson statistics would apply. But neither is the case in > nature. The variance in multiple electron-hole charge generation, called > Fano noise, is empirically described by > > $$ \sigma_{FN}^2 = F_F \eta_i = F_F \frac{h_v}{E_{e-h}} $$ > > where [$\sigma_{FN}^2$ is the variance of the Fano noise], and $F_F$ is > referred to as the Fano factor, which is defined by the variance in the > number of electrons generated divided by the average number of electrons > generated per interacting photon. [notation: $h_v$ and $E_{e-h}$ were > defined to be the photon energy in eV and the energy required to create an > electron-hole pair, and $\eta_i$ is the ratio of the two. So $\eta_i$ is the > expected number of electrons per incident photon, if all photon energy goes > into production of electrons.] Based on this description of the process, I would assume that each incident photon liberates an uncertain number of electrons, and therefore each incident photon should contribute a constant amount of variance. So I think the Fano noise variance should be **proportional** to the incident number of photons. But according to Janesick, the Fano noise contributes a constant variance **independent** of the number of photon arrivals. To be more mathematically precise, I think that the whole process should be modeled as a compound Poisson process, since both the number of arrivals and the number of electrons liberated per arrival are stochastic quantities. The wikipedia article on the compound Poisson process shows that the variance of the compound process has two components (one for the randomness of arrivals and one for the random size of the arrivals), and both components are proportional to the mean number of arrivals. So can anyone explain the discrepancy between my expectations and the book's equations? I feel I must have misunderstood what Fano noise is -- because if it's what I think it is, the natural model of it contradicts Janesick's equation. By the way, I know that $\sigma_{FN}^2$ is **not** referring to the variance per incident photon. If it were, then Janesick would multiply it by the mean photon arrivals when calculating the total image noise, but he doesn't. Here is his calculation of the total noise variance (equation 3.17, p. 34, irrelevant flat field term removed): $$ \sigma_{TOTAL}^2 = \sigma^2_{READ} + \eta_i F_F + \eta_i S $$ where $S$ is the expected number of electrons from the exposure. The first term is from shot noise, the second from Fano noise, the third is from shot noise. Note that the shot noise term scales linearly with $S$ but the Fano noise term is independent of $S$.


title: Does Heisenberg uncertainty apply within each quantum configuration, or in the amplitude distribution over them?

text: I'm still absorbing some basic ideas about quantum physics and now I think I have to reconsider the Uncertainty Principle. Here is what I understand, in summary:   * a "configuration" specifies the position, momentum, spin, and type, of every particle in the universe   * there is a complex amplitude distribution over the full space of configurations   * each configuration evolves into all possible subsequent configurations, and the associated amplitude transforms in some way for each of these   * the amplitude for a future configuration is the sum of amplitudes over all inbound evolutions   * this is basically what the Shrödinger Equation represents   * the probability that we are in any particular configuration is the relative squared amplitude. I think I get it. But now I've started to think about the Uncertainty Principle all over again. Long before I understood the above points, I understood that there _was_ no exact joint (position,momentum) value that could ever be measured. If I go back over it all now, I get the impression that, within each configuration, the position and momentum _is_ exactly defined and in fact the Uncertainty Principle comes from the fact that the _amplitude distribution_ itself is fuzzy. That is, if I make a measurement of momentum, I am projecting many configurations, encoding many velocities, onto a preferred basis. The information from the momentum measurement is "you are in one of these configurations, for which the corresponding position is one of these values". Is this interpretation correct? Follow-up question: does this imply that, if the amplitude distribution was pointlike, then uncertainty would be eliminated? (Not that this ever actually happens.)


title: Commutator not transitive

text: I noticed the following: $$[L_{+},L^2]=0,\qquad [L_{+},L_3]\neq 0,\qquad [L^2,L_3]=0.$$ This would suggest, that $L^2,L_+$ have a common system of eigenfunctions, and so do $L^2,L_3$, but $L_+,L_3$ don't. How is that possible?


title: Do alternate theories for Dark Matter (like MOND) explain it's effect on gravitational lensing?

text: For a long time, I was sceptical about the evidence for dark matter. To me, it seemed like a pretty big leap to make when we have no idea whether or not our current models of gravity _should_ apply exactly to cosmological objects of massive scales like galaxies. Just like Einstein’s relativity replaces Newton's laws, wouldn't MOND be a better explanation for the discrepancies of galactic rotations than some "dark matter" that we have no evidence of? Apparently though, it's effect on galactic rotation is not the only evidence of dark matter, we can also see the effect of dark matter on the gravitational lensing of galaxies. That seems a lot harder to explain using modified theories of gravity than the rotational problem. Do any of the modified theories of gravity address this evidence for dark matter, or just the galactic rotation problem?


title: Deriving 'frame dragging' in elliptical coordinates?

text: I'm currently trying to derive the equation that demonstrates 'frame dragging'. I am familiar (of sorts) with the Kerr metric, but I am most comfortable working in cartesian/elliptical coordinates? Is this still possible to derive, if so, can anyone help me? Thank you for your time! I have now derived that:$$ \Omega = \dfrac{r_s \alpha r c}{\rho^2(r^2 + \alpha^2) + r_s \alpha^2 r sin^2 \theta}$$ But what does $\Omega$ physically tell us?


title: Why positronium can annihilate in vacuum?

text: I thought that the annihilation process of positronium cannot take place without a third-party particle. This can be directly derived from energy & momentum conservation: energy conservation: $$h\omega=E_e+E_p+E_{third}$$ momentum conservation: $$\frac{h\omega}{c}\hat{\vec{n}}=\frac{\vec{V}_e E_e}{c^2}+\frac{\vec{V}_p E_p}{c^2}+\frac{\vec{V}_{third} E_{third}}{c^2}$$ Short transformation of these two formulas lead to: $$|\frac{\vec{V}_e E_e}{c}+\frac{\vec{V}_p E_p}{c}+\frac{\vec{V}_{third} E_{third}}{c}|=E_e+E_p+E_{third}$$ Where $h\omega$ is a sum over all created photons, $\hat{\vec{n}}$ is a unit vector. If there is no third particle, then this process cannot happen, because it would mean that electron and positron velocities had to be equal to speed of light. We cannot use vacuum fluctuations here, because vacumm balance must be preserved too (zero energy and zero momentum overall in vacuum fluctuations). I start to wonder if that can become possible if only higher decay modes are used: three or five photons in case of parallel spin of $e^{+}$ and $e^{-}$, four or six photons in case of antiparallel spin. But it seems impossible for me, because the $E_{third}$ term has to be massive otherwise would just cancel out from both sides (because if the created particle is an extra photon it has a speed of light, and $E_{third}$ terms are equal). It seems to me that some massive particle must be created or participate along the way. Maybe for example a neutrino and antineutrino (created) or just some neutrino (participate)? Maybe this is another example showing that neutrinos have mass? EDIT: well it appears that with your help we have found an answer. My original question ignored vectors. Now that the third formula includes a vector magnitude of momentum times c, the third particle can be a photon, and they cannot cancel out.


title: How do I develop a deeper understanding of physics?

text: I'm thinking about majoring in physics, but I don't think my background is really strong at the moment. I've completed the standard first year math and physics sequence (multivariable calculus/differential equations/linear algebra and University Physics by Sears and Zemansky). I want to utilize this summer before my second year to develop a deeper and more intuitive approach to both physics and math. How do would I start this process, and what books and other resources would be useful for a self-learner?


title: Should a radiation-filled Universe be scale invariant?

text: Imagine a spatially flat Universe, without cosmological constant, filled only with EM radiation. As Maxwell's equations without charges or currents are scale invariant then should this Universe be scale invariant as a whole? i.e. should there be no length scale associated with such a Universe? Friedmann's equation for such a Universe is conventionally given by: $$\left(\frac{\dot a}{a}\right)^2 \propto \frac{1}{a^4}$$ The solution is: $$a(t) = \frac{t^{1/2}}{t_0^{1/2}}$$ where $t_0$ is the current age of the Universe. Now let us calculate the particle horizon distance $D$ given by: $$D = \int_0^{t_0} \frac{dt}{a(t)}$$ We find that: $$D = 2 t_0$$ Therefore conventional theory says there is a length scale associated with this Universe. But is that reasonable? PS If for some reason we had $\rho \propto 1/a^2$ so that $a(t) = t/t_0$ then we could have a truly scale invariant cosmology where both the particle horizon and the cosmological event horizon diverge leaving no length scale.


title: What's wrong with my calculation of gravitational potential for a uniform sphere?

text: This is really embarrassing, but I'm not quite sure where I'm going wrong here... Why is this calculation of the gravitational potential inside a sphere with uniform mass distribution incorrect? **Set-Up** Let's say the sphere has mass $M$ and radius $R$ (and uniform mass density $\mu$), and what we want to find is the potential at any distance $r$ from the center of the sphere, where $r<R$. We normalize the potential to zero at infinity. **Calculation** The potential $\phi(r)$ is equal to the potential right outside of the sphere, plus the potential difference between some point inside the sphere and a point right outside. $$ \phi(r)=\phi_0-\int_R^r \frac{\mu G}{r}dV $$ (Sorry for using $r$ for the upper limit of the integral as well as for the variable in the integrand. Hopefully this doesn't cause confusion.) Now to figure out the different aspects of the above equation equation. The potential right outside of the sphere is: $$\phi_0=-\frac{MG}{R}$$ The differential volume element can be expressed as the constant-potential spherical shell's surface area times the shell's differential width: $$dV=4\pi r^2 dr$$ And one final detail, the mass density of the sphere: $$\mu=\frac{3M}{4\pi R^3}$$ Using this information, $$\phi(r)=-\frac{MG}{R}-\frac{3MG}{R^3}\int_R^r r dr$$ $$\phi(r)=-\frac{MG}{R^3}\left[R^2+\frac{3r^2}{2}-\frac{3R^2}{2}\right]$$ $$\phi(r)=-\frac{MG}{2R^3}(3r^2-R^2)$$ **Conclusion** This result disagrees with a few places I've visited, like this one, which states that the correct result (in terms of the variables I've used) is $$\phi(r)=-\frac{MG}{2R^3}(3R^2-r^2)$$ Both results give the same potential at $r=R$, obviously, but my result starts to look ridiculous for values like $r=R/2$. The only part in my calculation that seems sketchy to me is that first equation, where I talk about the potential difference at points inside and outside of the sphere; I don't know if it's correct to be dividing by $r$ in the integrand... Or maybe I just made a stupid algebra mistake somewhere in there. Where did I go wrong?


title: What happens to "large objects" within very small time periods?

text: What can we say about the state of "large things" within very small time periods? While reasonable (or "useful") divisions between classical and quantum physics are usually made in terms of size, can a similar division be made in terms of time? If so, what scale of time would that be? For example, suppose someone is "sitting at a desk". During very small time periods we cannot specify exactly where all the particles of the body are, right? Does that mean that the body (and everything else for that matter) is statistically "spread out" in small time periods?


title: Understanding EM Fields

text: I am an electronics engineering major and some questions arise when studying communications technology that utilizes wireless technology. In particular, I am more of a complete picture kind of person, as such I like to see how various things are connected. I want to understand, at least on the surface level, how the following things are connected: Phonons, photons, electrons, charge fields. As far as I understand things you have phonons. These are quasi-particles representing the quantized energy packets of vibrations through a lattice structure. This seems intuitive enough, and after some further reading I see that photons can be spontaneously emitted from photon interactions resulting from absorption and re-emission of the quantized energy of the phonons. I assume the photon emission frequency/energy is directly related to the energy of the phonon. I believe that photons can also cause phonons when a photon is incident to a lattice structure, though I am not 100% on the details of this. Photons are actual particles of quantized packets of energy propagating through the EM field medium. As to what exactly constitutes the EM field I am not sure. I believe magnetic fields are due to electrons? I also believe that electrons have an electric field due to the inherent charge electrons pose? However, I believe the charges reach/influence is based on the inverse square law. As such, is the electromagnetic field simply composed on the electron charges as the medium? If this is the case, then how can the EM field be so far reaching? I apologize for the large amount of questions, and I am sure there are similar questions around, but none that seem to tie all of these things together. EDIT: To add, am I correct to assume the EM Field is modeled as a continuous distribution of charge, as such the inverse square charge distance is a non- factor? And to answer one of the comments in the question, I have taken classes on partial differential equations, but my physics did not require them, thus I have only indirectly utilized Maxwell's Equations. I assume those will help connect a few of my questions mathematically, so I'll be sure to take a look at those.


title: Explanation: Simple Harmonic Motion

text: I am Math Grad student with a little bit of interest in physics. Recently i looked into the wikipedia page for Simple Harmonic Motion. Guess, I am too bad at physics to understand it. Considering me as a _layman_ how would one explain:   * What a Simple Harmonic motion is? And why does this motion have such an equation $$x(t)= A \cos(\omega{t} + \varphi)$$  associated with it? Can anyone give examples of where **S.H.M.** is tangible in Nature.


title: Common false beliefs in Physics

text: Well, in Mathematics there are somethings, which appear true but they aren't true. Naive students often get fooled by these results. Let me consider a very simple example. As a child one learns this formula $$(a+b)^{2} =a^{2}+ 2 \cdot a \cdot b + b^{2}$$ But as one mature's he applies this same formula for Matrices. That is given any two $n \times n$ square matrices, one believes that this result is true: $$(A+B)^{2} = A^{2} + 2 \cdot A \cdot B +B^{2}$$ But eventually this is false as Matrices aren't necessarily commutative. I would like to know whether there any such things happening with physics students as well. My motivation came from the following MO thread, which many of you might take a look into:   * http://mathoverflow.net/questions/23478/examples-of-common-false-beliefs-in-mathematics


title: How relevant is LHC to quantum gravity?

text: Premise: the LHC is obviously mapping unseen territory in high energies, and therefore it's always possible to imagine _far out_ results. Excluding completely unexpected outcomes - is the LHC performing any experiment that could help with string theory or m-theory? For example:   * direct super-strings or m-theory predictions to be tested or confuted but also   * measurements that would help "shape" string/m-theory into something more concretely testable or practical than the current blurry incarnation?


title: Knock out drum sizing for 10 times velocity reduction

text: How can I size a drum such that the velocity is reduced by 10 times inside the vessel. My velocity in the pipe entering the drum is 9.77 m/s. Velocity inside the drum should be 10 times less than 9.77 m/s![Suction Drum representation](http://i.stack.imgur.com/wRsnp.png)


title: Current in a circuit with resistors and capacitors

text: Suppose we have a circuit with both resistors and capacitors and some voltage source. If we want to calculate the current in the circuit, do we apply Ohms law? My doubt is that this would only take into consideration the resistors and not the capacitors, so do the capacitors play any role in the current flowing through the circuit?


title: Energy stored in a capacitor

text: Suppose we have a capacitor connected in series with a resistor and this is connected to a cell. So would the voltage across the capacitor be less than the the emf of the cell? And hence the energy stored? Because of the potential drop across the resistor?


title: The Dirac-Delta function as an initial state for the quantum free particle

text: I want to ask if it is reasonable that I use the Dirac-Delta function as an intial state ($\Psi (x,0) $) for the free particle wavefunction and interpret it such that I say that the particle is exactly at x=0 during time t=0? If I use this initial state, can I also use it to predict how the wavefunction should evolve in time? That is, if $\Psi (x,0) = \delta(x) $, then, $$ \phi(k) = \frac{1}{\sqrt{2 \pi}} \int_{-\infty}^{\infty} \delta(x)e^{-ikx} dx=\frac{1}{\sqrt{2 \pi}} $$ then, $$ \Psi(x,t)= \frac{1}{\sqrt{2 \pi}} \int_{-\infty}^{\infty} \phi(k)e^{i(kx-\omega t)} dk $$ or, $$ \Psi(x,t)= \frac{1}{2 \pi} \int_{-\infty}^{\infty}e^{i(kx-\omega t)} dk. $$ Is the above wavefunction a valid expression to see how a particle that is initially localized in the origin evolves in time?


title: Newtonian Physics vs Relativity - the results

text: Does anyone have examples of the results calculated by using Newtonian physics vs the same ones calculated using relativity, compared with real measurements obtained in those experiments? Please elaborate on the experiments and their circumstances too. Thanks. I am looking for numeric results ONLY and the experiments and circumstance around those results. I am not asking someone to explain the theory. Something similar to this would be a good start: ![relativity](http://i.stack.imgur.com/GHEMh.png) (source: http://en.wikipedia.org/wiki/Tests_of_general_relativity) Thanks.


title: Will porridge/rice absorb water faster without a lid on it?

text: I just wonder if my porridge would be ready any faster if I take the lid off my tupperware container.


title: What causes the Pauli exclusion principle (and why does spin 1/2 = fermion)?

text: It seems to be related to exchange interaction, but I can't penetrate the Wikipedia article. What has the Pauli exclusion principle to do with indistinguishability?


title: Relation of energy to amplitude & frequency

text: Not really a homework question per se, but i am just curious about this: Energy of a wave is given by the planck relation, which should be proportional to its frequency. However in electrical engineering we always learn that the energy of a wave is the square of its magnitude; nothing to do with frequency. So are these 2 concepts equivalent? Thanks.


title: Two spheres (A physics olympiad problem)

text: Browsing an archive of problems of a local physics olympiad, i stumbled upon a problem which seems not a very trivial. Given two identical metal spheres in vacuum, with mass $m$ and radius $R$. One sphere with a charge $Q$ and other with no charge(neutral). At initial moment, they are very far from each other. After releasing, due to the electrostatic attraction, the spheres come together and collide. Find the velocity of the spheres after the collision if the collision is: a) perfectly elastic   b) inelastic Ignore gravity forces and a possible spark discharge. The part b) seems especially tricky. **Edit:** Yes, there was probably a typo. The part b) must be simply "inelastic collision" that means kinetic energy is not conserved and the spheres do not stick. Sorry for confusion!


title: Can the effects of gravity be broken by jumping?

text: I was having a debate the other day with a work colleague where I explained that gravity is a weak force because it is easily broken. Then I remembered a lecture by someone, I forget who, that explained gravity is very weak because you can break its influence just by jumping or lifting a pencil, etc. He countered that with something along the lines of 'that even though the pencil or your body is being moved away from the source of gravity it is still affected by gravity and thus it has weight'. **Is jumping a good example of gravity being a weak force?** P.S. You can probably tell, my colleague and I are not physicists but we enjoy our little debates, we just need to get our facts straight.


title: Can someone explain the physics of a standing jump?

text: I would like to use something like a kinematic equation to know where the person will be in the jump at a given time. However since I think acceleration is not constant because of the jump force it doesn't seem I can use that equation. It seems like this is a summation of forces with a kinematic equation but it has been a few years since I've had to do this kind of thing. Any help is greatly appreciated!


title: quantization of this hamiltonian?

text: let be the Hamiltonian $ H=f(xp) $ if we consider canonical quantization so $ f( -ix \frac{d}{dx}-\frac{i}{2})\phi(x)= E_{n} \phi(x)$ here 'f' is a real valued function so i believe that $ f(xp) $ is an Hermitian operator, however How could i solve for a generic 'f' the eqution above ? , in case 'f' is a POlynomial i have no problem to solve it the same in case f is an exponential but for a generic 'f' how could i use operator theory to get the eigenvalue spectrum ?¿


title: Is the future already determined?

text: I've always wondered (and was re-inspired to explore further from these two videos) that if at a single point of time we know about the _complete state_ (position, momentum, spins, everything.) of every particle in an isolated system (e.g., the whole Universe), can we perfectly determine its future? And by same analogy, also the past? I understand that according to the uncertainty principle, it is impossible to determine all state properties of a particle accurately at the same time. So my question is, is it like future of an isolated system is already determined but is just not perfectly predictable by an observer because of limitations in observability? UPDATE: To be more clear, the question is **not** about whether we can or can not determine the future but about whether or not the future of an isolated system is already fixed. I.e., given exactly same initial conditions, will two isolated systems always be in same state with time? Is our future already fixed even though we'll never know it in advance? UPDATE: Would appreciate less technical and more accessible explanations. I'm not from Physics background but am interested in it nonetheless.


title: What are the thermal properties of a "perfectly rigid" container?

text: can a "perfectly rigid" container be a diathermal boundary? I checked the wikipedia article for heat and read: > Kinetic theory explains transfers of energy as heat as macroscopic > manifestations of the motions and interactions of microscopic constituents > such as molecules and photons. If the microscopic constituents of the container can't move (I assume this is what rigid means) is the only avenue for heat transfer across a rigid container the emission of photons?


title: Inertial frames

text: I'm just starting my study of relativity, and I have a rough understanding of the connection between inertial frames, newton's laws, and galilean transformations, but I'd probably benefit more if someone could spell out clearly what is taken as an assumption/axiom in classical mechanics (newtonian vs special relativity), and what is implied. I have a lot of loose information, and it would really help if someone could tie it all together. **I've heard that inertial frames are frames in which Newton's laws hold.** Now my textbook (classical mechanics, taylor), says that Newton's first law is implied by the second, and this first law is just used to determine which frames are inertial. So if an object doesn't suddenly accelerate with the influence of a force, you're in an inertial frame. So suppose the first law holds in a particular frame. How does it follow that the second and third laws also hold in that frame? **Wikipedia says that both newtonian mechanics and special relativity assume equivalence of inertial frames.** But what does "equivalent" mean in this context? **Any frame moving with constant velocity with respect to an inertial frame is also an inertial frame.** I know that if frame S is inertial and observe a force F, and if the respective force F' when viewed from S' (which moves at constant velocity with respect to S), the F'=F. This is stated as "newton's second law is conserved under a galilean transformation", but I'm not sure why. When demonstrating F=F', we assume F=ma in S and F'=ma' in S', so it seems like we assume the second law is true in both frames and simply show that F=F' Like I said, I know it's a lot of loose info, but I'd really appreciate it if someone could clarify/tie together everything


title: Why and how will the string break?

text: While i was searching on the identity discussed earlier that is $$1 + 2 + 3 + 4 + 5 + ... = -\frac{1}{12}$$ I found a similar identity applied to physics concept. It started by arranging long molecule modeled as a 1-dimensional string with alternating charges +q ,-q ,+q,-q and so one spaced evenly. The force between adjacent charges is known, it states that by pulling on one end of the molecule which is slowly increasing with time. One could determine the maximum force you can pull on the string so that the string does not break. and that it could be determined by using the identity $\sum _{ n=1 }^{ \infty }{ { (-1) }^{ n-1 } } \frac { 1 }{ { n }^{ 2 } } =\frac { { \pi }^{ 2 } }{ 12 }$ I really don't know how pulling on one end of a string will cause it to break somewhere along the string. Any ideas Thanks


title: Are states from an unpolarized source beam distinguishable or not

text: I have seen this pop up twice so far in my reading of Feynman Lectures III. For example, in the first equation of section 5-8 (it may take up to a minute for the equation to load), where the beam is ejected from a furnace and so states are randomly produced, the probablities are added rather than first interfering the probability amplitudes and taking absolute-squared. But I thought nothing was done to distinguish the states $+S, 0S, -S$. Moreover, in the situation of the fancy "extended" Stern-Gerlach apparatus where I split the beam into three beams according to the $S$ states but recombine them without blocking any of the split beams, I perform future calculations by interfering the three amplitudes. I am convinced here that I have to do interference. But why is the first paragraph different?


title: Mechanics question - ladder leaning against a wall

text: I am struggling with part a) of this question: ![question](http://i.stack.imgur.com/E4I44.jpg) Here is the solution: ![solution to question.](http://i.stack.imgur.com/XDe1M.jpg) In part a) they prove that $F \leq \mu R$ is false for this system. So as $F$ is neither less than nor equal to $\mu R$ it can only be greater than $\mu R$. I don't understand this. My understanding of friction is that the friction, $F$, can only increase up to the value of $\mu R$, it can't increase past this value. Whilst $F$ is increasing up to and equaling the value of $\mu R$ the system is in equilibrium. Then, after this point $F = \mu R$ but the opposing force overcomes this frictional force (i.e. "pushing force" $> \mu R$) and so the object moves and the system is no longer in equilibrium. Is my understanding not correct? Can friction be greater than $\mu R$? If this solution is incorrect could someone please tell me what's happening? I'm fairly confused at the moment.


title: Hamiltonian for the Periodic Kitaev Model

text: The Hamiltonian for a system of spinless fermions on a 1D chain (with chemical potential $\mu=0$) is given by $$ H=-\sum_j\left( c^\dagger_{j+1} c_j+h.c.\right)+\Delta \sum_j \left( c^\dagger_{j+1}c^\dagger_j+h.c.\right) $$ where $\Delta$ is some number. If we introduce $$ c_j=\frac{1}{\sqrt{N}}\sum_k e^{ikj}c_k$$ We obtain the result below: $$H=\sum_k \xi(k) c_k^\dagger c_k+\Delta\sum_k \left(e^{-ik}c_k^\dagger c_{-k}^\dagger +e^{ik}c_k c_{-k}\right) $$ where $\xi(k)=-2\cos(k)$I am trying to represent this Hamiltonian in matrix form by using the Nambu operator $$ \phi_k=\begin{pmatrix} c_k \\\ c_{-k}^\dagger \end{pmatrix} $$ Numerous texts give it as $$ H=\sum_k \phi_k^\dagger \begin{pmatrix} \xi(k) & 2i\Delta \sin(k)\\\ -2i\Delta \sin(k ) & -\xi(k)\end{pmatrix}\phi(k) $$ However, when I expand the above out, I do not get my original coupling term back--instead, I get $$ \Delta \sum_k \left( e^{-ik}c_k^\dagger c_{-k}^\dagger -e^{ik}c_k^\dagger c_{-k}^\dagger+e^{ik}c_k c_{-k}-e^{-ik}c_k c_{-k}\right) $$ I see that, to obtain my old coupling term, I have to let $e^{ik}c_k^\dagger c_{-k}^\dagger=e^{-ik}c_k c_{-k}=0$, but I can't explain why. Can someone please help me with this step? Here is a similar question posed in a problem set from a German university for your reference: http://users.physik.fu- berlin.de/~romito/qft2011/set6.pdf


title: Coordinate-free derivation of the Lamé-Navier's elasticity equations

text: Linear static elasticity provides a local equation $-\mathrm{div}\sigma=f$, the constitutive law $\sigma=2\mu\epsilon+\lambda \mathrm{tr}(\epsilon)I$ as well as the strain-displacement relationship $\epsilon=1/2(\mathrm{grad}u+\mathrm{grad}^T u)$ (boundary conditions are ignored) where:   * $\sigma$ is the second-order stress tensor   * $\epsilon$ is the second-order strain tensor   * $f$ is the vector of external distributed force   * $I$ is the second-order identity tensor   * $u$ is the (space-dependent) displacement vector   * $\mu$ and $\lambda$ are Lamé's coefficients The Lamé-Navier version of the local equation is solely expressed in terms of vector $u$ as follows:$$ \mu \Delta u+ (\lambda+\mu) \mathrm{grad}(\mathrm{div}u)+f=0$$ Derivations found in books all use a Cartesian coordinate system to derive this identity which is then re-expressed in an "intrinsic" coordinate-free vector form. Is there a way (or a reference) detailing how to derive this identity directly in a coordinate-free fashion?


title: Terminal velocity for rotating system

text: Suppose we have a small bob/ball attached via rigid rod to a pivot in a viscous medium and then rotated by a constant torque, will the system finally(after considerable time) rotate with constant velocity, or will the viscous torque not be able to completely cancel out the applied torque and the system will be accelerated forever ? If the viscous torque will cancel out the torque, is there a method to calcupate this viscous torque ? As well as the terminal angular velocity. Also, if we attach the pivot to a dynamo, and the same constant torque is applied on the system, will it still rotate with the same terminal angular velocity ?


title: Circular motion and time dilation

text: Einsteins' time paradox was shown when two atomic clocks were synchronized and one was sent in a super sonic plane and the other kept at rest. When the two where then checked it was shown that yes indeed the clock that took the plain trip had lost time when compared to the clock that was kept at rest. My question is what if instead of one of the clocks going on a plain it was put into a centrifuge or some other device that spun it around and very high speeds would it feel the effects of time dilation in the same manner?


title: Special Relativity Time problem

text: Suppose there are two observers, they each own a light clock which were synchronised in the beginning and are moving relative to each other. So each of them accuses that the other person's clock is running slow. Now they see a concrete event, like someone clapping or a bulb shining and note down the reading on their clock and send each other the information, now if the two pieces of information were different, which means one of the clock was actually running slow, so they will know who was moving actually. If the readings are same, then how will you justify Lorentz-Transformations for time ?


title: How does the pressurization in a house's water supply allow adjustable flowrates from a tap?

text: John Rennie's answer to this question got me thinking about the water supply in a house. I know that water supplies are normally operated at an over- pressure with respect to atmospheric pressure to make sure that if you open the tap the water will flow out due to the lower pressure at the exit of the tap (including a correction for pressure losses due to the plumbing and potential height differences). **What I am wondering is how adjusting the tap results in an adjustment of the flow rate from that tap if the water pressure in the main supply is a constant (probably by approximation)?** Because in that case the pressure upstream is fixed, the pressure downstream (atmosphere) is fixed so pinching or opening a valve should not change the flow rate. Obviously it does change the flow rate so there must be something else going on. The only thing I can think of is that the water supply somehow acts as if there are parallel connected (hydraulic) resistances as illustrated in the schematic below which makes sure that the distribution at the flow split can be adjusted. Is that the right thinking or is there a different argument why the flow is adjustable? ![enter image description here](http://i.stack.imgur.com/UZnLe.png) (The dashed line indicates some way of rerouting the water back to the mains, although that cannot be a direct connection because that would allow a pressure drop over this section)


title: Can dark matter be relativistic dust?

text: As far as I know the mass of an observed object increases as it approaches the speed of light. Is it possible that the excess mass called "dark matter" is due to relativistic dust? Surely, stars cannot account for that excess mass since we can measure their speeds and they are not relativistic. However, can we observe and measure the velocity of dust in the galaxy? Maybe the dust could account for missing matter...


title: Can someone help me understand this simple derivation for hall voltage?

text: I have a question regarding the derivation for hall voltage as shown in this image: ![enter image description here](http://i.stack.imgur.com/PlF3Q.png) How did they get $\Delta V_H = E_H d$ term? ($d$ is the width of the conductor)


title: Calculation of the magnetic field of a solid charged planet-sized sphere

text: How do I calculate the magnetic field of Mars, which is an almost solid sphere ? I'm a beginner but i'm currently doing a project on Terraforming Of Mars. To create an artificial magnetosphere on the planet, how do I calculate the magnetic field necessary? Please help me.


title: Show that the plane of incidence is perpendicular to the surface of reflection

text: Is it possible to derive from the boundary conditions of the Maxwell equations for E and H, that the plane of incidence for an EM wave is perpendicular to the reflection surface? How? If not, what additional condition is needed?


title: Chemical potential in Thermodynamics

text: In many scenarios, on computing the partial derivative of the internal energy (U) with respect to mole number (N) is negative. This implies that adding more moles of the substance decreases the U of the system. I find this strongly counter-intuitive. Shouldn't the derivative always be positive?


title: How to calculate energy in two-band Hubbard model

text: It might be a very easy question for you, but I am confused and I need helps. In the simplest Hubbard model at one-dimensional lattice, I ignore the $U$ term and only remain the hopping term. $$H=-t\sum_{<ij>}{c^{\dagger}_{i} c_{j}}$$ where $c^{\dagger}_i$ and $c_i$ are creation operator and annihilation operator on site $i$, respectively. And the Fourier transformation is defined as $$c_{k}=\frac{1}{\sqrt{V}}\sum_{i}{c_{i} e^{i k r_{i}}}$$ here, the summing index $i$ is subcript, and it is different from the imaginary unit $i$. Finally, we get $$H(k)=\sum_{k}{\epsilon_{k} c^{\dagger}_{k} c_{k}}$$ by sum over all site index $i$, and $<ij>$ gives geometric factor which is included in energy $\epsilon_{k}$. \-------o-------o---------------o-------o----                   |<-a->|<----b---->|                   bilattice      Here, my qusetion is, for bilattice, $\epsilon_{k}$ should give two energy bond -- one $k$ should give two differnet energy. I don't know how to calculate it. Could you help me?


title: Color factor for squark-quark-antiquark vertex

text: I am trying to do a tree level calculation of cross section in a process that involves sbottom exchange. There is a $ \widetilde{b} q \bar{q} $ vertex, where $q$ and $\bar q$ are quark and anti-quark of the same flavor. I have following questions:   1. How is the color conserved in such vertices? The problem is that (s)quarks carry only one unit of color.   2. What will be the color factor from this vertex if written explicitly? (Just like Griffiths does in his book) PS: My knowledge of QFT, QCD, and SUSY is very limited, almost none. I got the vertex factors and propagators from the superpotential for MSSM without R-parity using the method outlined in Griffiths' book. My answer is correct upto a color factor, hence I know THAT is the source of the problem.


title: Analogy between fluid dynamics and electromagnetism

text: I've been reading a set of notes (Chapter 13 of Caltech Ph 136 notes from 2004, by Blandford and Thorne) that draw an analogy between fluid dynamics and electromagnetism, identifying the magnetic vector potential $\mathbf{A}$ with the velocity flow field $\mathbf{u}$, the magnetic field with vorticity $\mathbf{\omega}$, and the electric field with $\mathbf{\omega} \times \mathbf{u}$. This is different from the hydraulic analogy for electric circuits, which I think identifies the velocity field with current density. Several things go through this analogy in a reasonable way, like $\nabla \cdot \mathbf{B} = 0$, and incompressibility corresponding to the gauge choice of $\nabla \cdot \mathbf{A} = 0$. But it looks to me like this analogy is not an "isomorphism". For example, does the condition $\mathbf{E} = \mathbf{B} \times \mathbf{A}$ (which follows, in the analogy, from the definition of vorticity) have any interpretation in electromagnetism? If our flow is irrotational, what does the velocity potential correspond to (e.g. does a scalar potential for the magnetic vector potential ever come up/have a name)? I'd be interested in any reference that thoroughly explores this analogy. The best I've found is here, but as far as I can tell it does not address the above issues (and similar).


title: What happens when an electron jumps through the event horizon of a black hole?

text: If quantum teleportation is performed into a black hole (by an electron for example), what happens to that electron? Let's say a hydrogen atom is very close to a black hole and the electron jumps into the next level of energy, which collides with the black hole, is the electron stuck forever? Is it prevented from entering the black hole? Is it free to enter and leave? **EDIT** [More precision about my train of thoughts, feel free to either answer or discuss about my why of thinking if not appropriate for the concern] : I understand that we do not know enough about the QFT and black hole interaction, but I would like to try to narrow a bit the question, to see if there is a way to actually answer the question with a relative precision. Theoretically, at the moment the electron jumps and his electron cloud conflicts with the black hole, I believe the only possibilities are :   * The electrons "enters" the black hole at one point (quantum teleports its state into it) and is free to leave the part of the cloud that is in the black hole, resulting in a paradox in the fact that no information can enter then leave the black hole. _So not possible ?_   * The electron does NOT enter the black hole and is trapped into a electron cloud which is a part of the real cloud, resulting in a disorder in the electromagnetic stability of the atom, and making the electron "drop" into the atom. (what could that imply? is it possible ?)   * The electron enters the black hole, and is trapped in it, breaking the link between the nucleus and the electron. ( what could that imply ? Is it possible ? Breaking the link requires energy, so the black is loosing energy/mass, while also earning energy because of the mass gained by the absorption ?). But that would mean that the nucleus "sees" the electron entering the black hole because of the link broken or would that mean that the nucleus still considers itself a hydrogen atom ? I know this is very theoretical, and my train of thoughts is probably really bad, but I found it interesting to think about.