title: Does Galileo's Tower of Pisa argument contradict quantum mechanics?

text: (My questions are at the end, but they may not mean much without explanation below.) Galileo argued that because the mass of a falling object can always be redistributed in ways that asymptotically approximate a set of smaller falling objects, e.g. linked by light strings, then the equations of motion for an object falling in a vacuum cannot depend on its mass. That is, because dependency on mass would lead to paradoxical motion predictions if in one case you used the total mass $m_\Sigma$ and in another one of the asymptotically free submasses $m_i$ were used as the arguments to the equation of motion. Thus for example, the equation for the instantaneous velocity of any object falling in a vacuum does not include mass as a significant parameter: > $v_i=\sqrt{2gd}$ Galileo's argument was insightful but largely tautological. The problem is that the equations of motion for objects falling though fluids, such as terminal velocity, provide an existence proof that self-consistent equations of motion with mass as a significant parameter _can_ be constructed. However, such self-consistency becomes possible only if the distribution of object mass in space also becomes a non-trivial parameter. This additional information allows paradoxes to be resolved by converting seemingly inconsistent state predictions into asymptotic limits of a larger continuum of predictions. For example, in fluid mechanics a mass $m_{\Sigma}$ is correctly predicted to have a higher terminal velocity than the same mass configured as two smaller spheres $m_1$ and $m_2$ linked by a thin string. The latter form is an example of how the predicted behavior of a single falling sphere can be made to approach asymptotically the prediction for two smaller spheres by slowing "morphing" the original object into one that closely approximates the case of two smaller spheres. What Galileo really proved, then, was that since experimental evidence showed that mass was not a relevant parameter for the equations of motion of objects falling in a vacuum, then the specific distribution of mass in any one object must also be irrelevant to those equations. What Galileo could never have realized was that exists a domain of physics for which his argument is very relevant: quantum physics. The de Broglie wavelength $\lambda=h/p $ of an object falling at modest speeds in a vacuum is a function of its instantaneous momentum $p_i=mv_i$, so: > $\lambda_i=\frac{h}{mv_i}=\frac{h}{m > \sqrt{2gd}}=\frac{h}{m}\sqrt{\frac{1}{2gd}}$ While not an equation of motion, this equation violates Galileo's argument by predicting that the de Broglie frequency of an object falling in a vacuum _is_ a function of its mass. As in the case of fluid dynamics, the inclusion of mass as a significant parameter results in paradoxical predictions unless and until the configuration of the mass in the object is also taken into account. In other words, the de Broglie wavelength of an object must necessarily depend upon its shape and how mass is distributed within that object. The resulting more closely resembles solid state acoustic resonances in complexity than it does the simple singular mass parameter of pointlike particles isolated in space. For example, if $m_1=m_2=m$ and $m_\Sigma=m_1+m_2=2m$, three state equations are relevant: > $\lambda_{\Sigma}=\lambda_{2m}=\frac{h}{2m}\sqrt{\frac{1}{2gd}}$ > > $\lambda_1= \lambda_m=\frac{h}{m}\sqrt{\frac{1}{2gd}}$ > > $\lambda_2= \lambda_m=\frac{h}{m}\sqrt{\frac{1}{2gd}}$ Self-consistency alone argues that if these equations are even approximately correct, then $\lambda_{2m}$ should dominate when the $2m$ mass takes the form of a compact spherical object. Conversely, reconfiguring the $2m$ sphere into two smaller $m$ spheres linked by a slender thread should asymptotically approach the case of two separate $m$ spheres, and so should be dominated by the $\lambda_m$ wavelength. However, any single wavelength cannot be accurate in any of the cases. There should instead be a spectrum of wavelengths and intensities that is highly dependent on the particulars of how the mass is distributed and connected in the object. That is why I mentioned earlier that an accurate equation model for predicting de Broglie wavelengths for non-point object will necessarily be comparable in complexity and subtly to resonance models in solid state physics. I do not readily see a way out of this. Galileo's argument, while largely specious in the case of the equations of motion for objects falling in a vacuum, is extraordinarily relevant to quantum mechanics, where it appears to say that the entire concept of "mass" requires revisiting if it is to be used accurately and meaningfully. A correct model should instead have more kinship with the equations of motion for objects falling though a fluid, in the sense that both must define very carefully how the distribution and "connection" of mass impacts the predictions. * * * So, my two main questions are:   1. Do there currently exist mathematical models of de Broglie wavelengths that pass the Galilean self-consistency test? For example, are they perhaps hidden away in the mathematical details of phenomena such as quantum models of molecular vibration? (My impression is "no", but I could certainly be wrong.)   2. What are the particle physics implications, if any, of quantum mass being dependent on form?


title: The fundamental equations of electromagnetism

text: I'd like to know what are the basic equations of electromagnetism, that can be used to formulate all the other laws and equations. Those basic equations I can think of are Maxwell equations, Lorentz force equation and Coulomb's law. What are the other fundamental laws and equation?


title: Why doesn't the air in a Standard Diving Dress crush you?

text: I recently watched a Mythbusters episode where they tested decompression in a tethered diving suit (which I believe is a Standard Diving Dress). When they removed the air pressure, the suit crumpled up and forced the body into the helmet. They had the suit 300ft underwater, and said that was something like 135psi. My question is, why wasn't the body crushed by the air pressure keeping the suit inflated? If the air needs to exert 135psi to keep the suit inflated against the water, why does that not crush the person in the same way the water does? I know that in this case they made a body which isn't "airtight" in quite the same way that a human is, but live people use these suits as well.


title: Diamagnetism of a degenerate electron gas for weak fields

text: In the book "Statistical Physics, Part I ($3^{{\rm rd}}$ edition)" by Landau and Lifshitz, at $\S59$ when he treats the diamagnetic part of the magnetisation of a degenerate electron gas for weak fields ($\mu_BH\ll\varepsilon_F$, where $\varepsilon_F$ is the Fermi energy), he says that the energy levels of the orbital motion are given by $$ \varepsilon=\frac{p_z^2}{2m}+(2n+1)\mu_BH,\;n=0,1,2,\ldots $$ where $p_z$ is the component of the momentum in the direction of the field $H$ and $\mu_B=e\hbar/2m_ec$ is the Bohr magneton. That's ok! But then he says that the number of states of the particle, at a given value of $n$, such that its momentum lies in the interval $p_z$ and $p_z+{\rm d}p_z$ is $$ 2\frac{VeH}{(2\pi\hbar)^2c}{\rm d}p_z,$$ where the factor $2$ accounts for the two possible spin orientations. I tried to reach this result, but I had no success. And that's strange, because the number of states depends on the field $H$. Well, if you could help me understand this, I would be very grateful.


title: Grounding insulative material

text: When grounded, can an insulative material keep its charge for any measurable length of time? Or, I suppose, if it was a perfect insulator, would it discharge at all? An example might be a charged rubber tire placed on the ground.


title: Are there two different ways to express the position operator $x$ in terms of the creation and annihilation operator?

text: As we known, to express the position operator $x$ in terms of the creation and annihilation operator $a^{+}$ and $a$, one way is:   $$x= \sqrt{\frac{\hbar}{2\mu\omega}}(a^++a);$$ $$p= i\sqrt{\frac{\mu\hbar\omega}{2}}(a^+-a).$$ But how about $$x= -i\sqrt{\frac{\hbar}{2\mu\omega}}(a^+-a);$$ $$p= \sqrt{\frac{\mu\hbar\omega}{2}}(a^++a)? $$ I want to know whether the two expression are fine. PS: It seems that I can use them to calculate something like fluctuation $\langle\Delta_x\rangle^2$ at coherent state $|\alpha\rangle$. Both expressions can give the right answer. By the 1st expression, $\langle\alpha|x^2|\alpha\rangle=\frac{\hbar}{2\mu\omega}[(\alpha+\alpha^*)^2+1]$, $\langle\alpha|x|\alpha\rangle^2=\frac{\hbar}{2\mu\omega}[(\alpha+\alpha^*)^2]$ then, $\langle\Delta_x\rangle^2=\langle\alpha|x^2|\alpha\rangle - \langle\alpha|x|\alpha\rangle^2 =\frac{\hbar}{2\mu\omega}$. By the 2nd way, $\langle\alpha|x^2|\alpha\rangle=-\frac{\hbar}{2\mu\omega}[(\alpha-\alpha^*)^2-1]$, $\langle\alpha|x|\alpha\rangle^2=-\frac{\hbar}{2\mu\omega}[(\alpha-\alpha^*)^2]$ therefore, $\langle\Delta_x\rangle^2=\langle\alpha|x^2|\alpha\rangle - \langle\alpha|x|\alpha\rangle^2 =\frac{\hbar}{2\mu\omega}$. The result is the same. And if we check the dimension, this formula is also OK.


title: Relative motion question

text: I'm sitting on my seat in a train and the train is moving at a very high speed, let's say 600 mph. If I throw a ball vertically up in the air (while still sitting on my seat), will the ball fall back in my hands or will it fall in the seat behind me since my position would have changed from the time I threw the ball, but the balls vertical trajectory will remain the same?


title: Ball thrown in a moving train

text: > A ball is thrown upward in a train moving with a constant velocity. Where > will it land? My intuition tells me that the ball will fall at my back. But my book says that it will return back to the thrower.


title: Heisenberg XXX time evolution operator for three qubits

text: I've a problem to reproduce the result in equation (4) on page three of this paper: http://arxiv.org/abs/0802.2588. So far I've understood that they apply a Heisenberg XXX interaction between 2&4, 4&6, 1&3 and 3&5\. After this they measure with $\frac{1}{2}(|0000><0000|+|1111><1111|)$ and calculate the fidelity. I think I calculate the last two steps correctly but I'm not sure about my implementation of the Heisenberg-Interaction. At the moment I apply four matrices $U(\alpha)\otimes 1\otimes 1\otimes 1\otimes 1$, $1\otimes U(\alpha)\otimes 1\otimes 1\otimes 1$, $1\otimes 1\otimes 1\otimes 1\otimes U(\alpha)$ and $1\otimes 1\otimes 1\otimes U(\alpha)\otimes 1$ one after another (1 for the neutral element) after I sorted the qubits in the right order with swap gates. That means the initial state is $\rho_F\otimes\rho_F\otimes\rho_F$ with order "1-2-3-4-5-6" and I sort them to "1-3-5-2-4-6". But that doesn't seem right. Or is my measurement projector wrong? Edit: Maybe I should add what I mean by $U(\alpha)$: When you calculate the time evolution operator from the Heisenberg Hamiltonian you get a term $\int dt J(t)$ in the e-function where J is the strength of the interaction. It depends on $t$ because you turn the interaction `on and off'. Now set $\alpha=\int dt J(t)$. That is $U(\alpha)$ just means time evolution is created by turning the interaction on and off. Greets


title: What is $k$ in Newtons Law of Cooling?

text: I am writing the definition of quantities for Newton's Law of Cooling and am not sure what to actually call k$,$ or its units. This is what I have so far $$\frac{dT}{dt}=-k(T-T_s)$$ Where $\frac{dT}{dt}$ is the rate of cooling measured in degrees Celsius per second $\left[°C\cdot{}s^{-1}\right]$ $T$ is the temperature of the object measured in degrees Celsius $\left[°C\right]$ $T_s$ is the ambient temperature of the objects surrounding measured in degrees Celsius $\left[°C\right]$ $k$ is a constant and has no units? I don't think that's a good enough description


title: Magnetic susceptibility in 1/eV

text: In this paper the authors refer to > transverse susceptibility $\chi_{ \perp}$ [meV $^{−1}$] I was taught that the magnetic susceptibility is dimensionless. How do I get $\chi$ in the above units??


title: Plasma Question: Expression for electron temperature in terms of the voltage of each prong of a triple Langmuir probe?

text: So I am trying to derive an expression for electron temperature based on the voltages on a triple Langmuir probe. I have pretty much been following the triple Langmuir probe derivation on wikipedia but I cannot reproduce their result. I am wondering if I am missing something. I get confused when they go from $\frac{1}{2} = \frac{1-e^{-e(V_{fl}-V_{+})/K_{B}T_{e}}}{1-e^{-e(V_{-}-V_{+})/K_{B}T_{e}}}$ To, $(V_{+}-V_{fl}) = \frac{k_BT_e}{e}\ln 2 $ using the limit $eV_{Bias} = e(V_{+}-V{-}) >> k T_{e}$. When I try to apply this limit to the first equation that I have here I just find that the denominator goes to zero which doesn't seem very helpful to me. For example what I thought was that $e(V_{-}-V{+})$ was equal to $-eV_{Bias}$ so I could multiply both sides of the limit by $-1$ which gives, $e(V_{-}-V{+}) << -K_{B}T_{e}$. So my exponential equation in denominator on the right side of the original equation would become: $e^{\frac{e(V_{-}-V{+})}{-K_{B}T_{e}}}$, which is basically equal to $1$ from the limit. So then in the original equation the denominator would become $1-1$ which is $0$. So I am pretty confused about this right now and any help that would point me to the correct derivation for the electron temperature on a triple Langmuir probe would be much appreciated.


title: Does the direction of perceived time depend on location in the universe?

text: Do the laws of physics allow for the possibility that in a galaxy far far away, the perceived arrow of time is in the opposite direction to ours, where people (aliens) die before they are born, and from our perspective appear to be living through time in the wrong direction?


title: The visibility of air

text: For pilots of gliders or sailplanes, the 'thermal' is the most important phenomena of the air. A thermal is classically described as an upward flow of air caused by ground level heating of air that rises in bubbles or a connected stream of warmed air. Given sufficient velocity of the rising air, a gliding craft, bird or even trash and debris can be lifted thousands of feet. It can be also noted that 'dust devils' can result from especially vigorous flows and that other even stronger phenomena like tornadoes and cyclonic storms are related. But in the absence of markers like dust, the air flow is generally invisible until possibly the flow reaches an altitude where the water vapor contained, condenses to form a cloud. And again if the flow is strong enough and contains enough water, a thunder storm is possible. So, the question. Given all of the above is it possible to see the mass of air that makes up the thermal? Is there anything in the difference between the thermal and the surrounding air that could be detected and presented graphically?


title: Practical personal aircraft today?

text: Is it possible to build, today, a personal aircraft that not use an impracticable amount of fuel for everyday use? What are the physical concepts that could be used to build it?


title: Momentum and indirect bandgap

text: When an electron is excited from valence band to conduction band, it has to have a finite momentum in the case of indirect bandgap. Does that mean that the electron cannot be created at rest ? does it have a non zero "speed" ?


title: What is "Quantum Levitation"?

text: I just found this video Controlled Quantum Levitation on a WipeOut Track  and I'm having a hard time finding the term "Quantum Levitation" used except in reference to the video. What is the proper name for this phenomenon and is this a real application of it?


title: Modified Coulomb potential

text: I'm working through Byron and Fuller's "Mathematics of Classical and Quantum Physics" and came across this problem: If the electric potential of a point charge were $\phi(r) = \frac{q}{r^{1-\epsilon}}, \epsilon \ll 1$ instead of $\frac{q}{r}$, many of the results in physics would be altered. Compute $E = -\nabla \phi, \nabla \cdot E, \nabla \times E$ and derive the analog of Gauss's Law for a spherical region of radius R. Examine the limit as $\epsilon \rightarrow 0$ and compare with the usual Coulomb potential. I worked them out and while the answers are different the physics seems essentially the same, and reduces to the normal answer as $\epsilon \rightarrow 0$. I'm trying to find more information about this problem. Thanks!


title: How to determine how many atoms are in something?

text: I want to know if it's possible to find out how many atoms are in something, and how one would do that. For example, how many quarks are in my brain(easy to find out once you know how many atoms there are)? So I ask again ... how does one find out how many atoms are in something, and of what kind of atoms these are? EXAMPLE: My brain.


title: Error in Chapter 15 (Mechanical Waves) of University Physics with Modern Physics - 13th Edition?

text: After reading this question I figured someone here would be able to answer this as well: I thought the answer to part (c) of this question: ![University Physics with Modern Physics \(13thE\) - pg 482](http://i.stack.imgur.com/1dgiS.jpg) (Figure 15.8 to the right:) ![University Physics with Modern Physics \(13thE\) - pg 477](http://i.stack.imgur.com/p0hHj.jpg) was $$\frac{3}{8}T$$ but in page 508 it's written: ![University Physics with Modern Physics \(13thE\) - pg 508](http://i.stack.imgur.com/DwnnI.jpg) I can see the particle at C has a downward acceleration at $t=\frac{5}{8}T$, but does it have an **upward velocity**? Thanks in advance for the attention.


title: Work done by gravity on a ball & the ball on earth

text: I have studied today that if a ball was to fall a certain height, then the work done by gravity on the ball would equal the work done by the ball's equal and opposite gravitational pull. By $W=Fd$, this means that the earth must travel the same distance as the ball. So if the ball falls 300 meters, the earth would eventually rise 300 meters in the direction of the ball's gravitational pull? Can you please give me your input on this particular matter because I don't understand it well ?


title: Power dissipated on internal resistance of short-circuited voltage source

text: Suppose we have a voltage source with an EMF of $\mathcal{E}$ and an internal resistance $R$. If we connect to it a perfect wire with zero resistance, we get a short circuit. The value of the current is determined by: $$I=\frac{\mathcal{E}}{R+0}=\frac{\mathcal{E}}{R}$$ Then the voltage across the source would be $V=\mathcal{E}-IR=\mathcal{E}-\mathcal{E}=0$ volts. Now I want to find how much power is dissipated do tue the current inside the source (e.g. Joule heating inside a battery due to collisions of electrons with ions). The formula is pretty simple: $$P=VI$$ So for this example $P=0 \cdot \frac{\mathcal{E}}{R}=0$ But we can rewrite the formula using Ohm's law: $$P=VI=I^2 R$$ And then it would be $P=\frac{\mathcal{E}^2}{R} \neq 0$ We get here a contradiction. Where is my mistake here? Maybe I have a wrong understanding of voltage? Is it because we have an EMF i.e. some non- electrostatic force which does work despite the fact we have short-circuited the source? The voltage itself as I understand describes an electric field which is not present. So maybe we should use instead of $V$ in Joule's heating formula an emf? But then, it still confuses me a little, because the formula itself is derived from Ohm's law and the electrostatic fields.


title: How is Space-Time curved?

text: How does space-time curved by mass/energy if there's nothing to be curved? I haven't seen any satisfying answer.


title: Why can charges outside be ignored in Gauss's Law?

text: In MIT's 8.02 course, it is shown in lecture 3 that we can derive Gauss's Law from Coulomb's to get $ \phi = \oint \vec{E} \cdot \vec{dA} = \frac{Q_{enc}}{\epsilon_{0}} $ However, in the lecture, it was assumed that there were no charges outside the surface. Later, Gauss's Law was used and the charges outside the sphere were ignored. I've been thinking about it and came across Can someone give an intuitive way of understanding why Gauss's law holds?. The first answer helped a bit. I think that the lecture's argument still holds when there are charges outside because $ \vec{E} $ is added vectorially. Suppose we have $ Q_{enc} $ in a sphere and $ Q_{out} $ outside the sphere. Then, By definition, $ \phi = \oint (\vec{E_{Q_{enc}}} + \vec{E_{Q_{out}}}) \cdot \vec{dA} $   $ = \oint \vec{E_{Q_{enc}}} \cdot \vec{dA} + \oint \vec{E_{Q_{out}}} \cdot \vec{dA}$   $ = \oint \vec{E_{Q_{enc}}} \cdot \vec{dA} + 0 $   $ = \frac{Q_{enc}}{\epsilon_{0}} $ This assumes that $ \oint \vec{E_{Q_{out}}} \cdot \vec{dA} = 0 $ because any field lines that go in must come out (hand wavy but I'm just going to accept it for now). I'd just like to know if this argument is sound.


title: How to prove that a hamiltonian system is not integrable?

text: To show that a system is integrable, we just need to find $N$ independent functions $f_j$ such that $\\{ f_i, f_j \\} = 0$. But how to prove that such a set of functions do not exist? For example, how to do this for the three-body problem?


title: How do I find the perpendicular velocity of a particle to a varying magnetic field?

text: I am trying to find the component of velocity perpendicular to a magnetic field. This was easy when the magnetic field was static and pointing in only one direction (the $z$ axis), but now I need to find it for any arbitrary velocity and magnetic field vector (in Cartesian coordinates). I have tried subtracting the dot product of the two vectors from the magnitude of the velocity vector, but that doesn't seem to be working.


Why is Erbium Doped Fiber Amplifier (EDFA) preferred over Erbium Doped Waveguide Amplifier (EDWA)? The question has been asked from an engineering point of view, but obviously I would also be interested in Physics of it. Is it just because of the difficulty of constructing a waveguide, or is there something more? I understand the individual benefits and limitations of a fiber amplifier and a waveguide but cannot understand the preference in this case. http://en.wikipedia.org/wiki/Optical_amplifier#Doped_fiber_amplifiers


title: Why is the photon emitted in the same direction as incoming radiation in Laser?

text: When an atom “lases” it always gives up its energy in the same direction and phase as the incoming light. Why does this happen? How can this be explained? How does the photon generated because of stimulated emission, know which direction to take? What are the factors leading to this?


title: Would a light or a heavy ball roll fastest down a slope?

text: A small, light ball and a larger, heavier ball are released from the top of a slope. Which will move further? which will come down faster?


title: Equivalent of the Kirsch Equations for an elliptical rather than circular hole

text: Is there an equivalent set of closed form equations to the Kirsch Equations but for an elliptical hole rather than a circular hole?


title: How To Use Ladder Operators?

text: I'm studying for a test in quantum mechanics and I'm having a hard time understanding how to use ladder operators. There are no examples in my text book, only definitions that I can't understand how to use, so I hope you can help me instead. So, there's an assignment where you're supposed to calculate first order correction to the ground state using ladder operators. In the right answer they say that $$E_0^1 = \langle 0|H'|0\rangle$$ where does the zeros come from? I understand that they're coming from the wave function $\psi$. But I don't really understand the theory behind this. Then, after some calculations it's written in the right answer $$ \langle 0|aa^\dagger a^\dagger|0\rangle +\langle0|aaa^\dagger a^\dagger|0\rangle =\langle0|aa^\dagger a|1\rangle +\langle0|aaa^\dagger|1\rangle$$ What are the mathematical rules of these ladder operators? How did these zeros become ones? I would really appreciate if someone could help me understand this a little bit better.


title: To calculate pressure at point in a vertical water pipe of given diameter

text: Is it possible to calculate pressure at a point in a pipe, where a solenoid valve shall be placed, given that   * the pipe carries water (utility supplied, regular water)   * from an overhead tank (at a certain height from ground)   * height of tank (i.e. max water column height, although not sure if vertical pipe-length should be added to it ?)   * water is flowing purely under the effect of gravity (and atmospheric pressure)   * value is installed at a certain height from ground   * pipe material is PVC   * pipe diameter is say 1.5inches.


title: How to understand dynamics $\dot x_i=\partial_jA_{ij}$ from skew-symmetric potential $A$?

text: We speak of a dynamical system with a potential if there is a scalar possibly depending on coordinate such that the vector field is exactly the (negative) gradient of the potential. That means each point moves along the direction with the steepest gradient descent. Also we know that the direction of gradient is perpendicular to level set. All of these are about understanding of dynamics in terms of potential. But how about vector potential? Given a vector potential $A$, what can we say about the system $\dot x=\nabla\times A$? I have no idea what **geometric intuitions** can be drawn from $A$. Even I don't know why we call it a potential if there's nothing similar to scalar potential besides something to do with $\nabla$. Similar questions I have for $n$ dimension where $A$ is a second-order skew- symmetric tensor and the system is $\dot x_i=\partial_jA_{ij}$. Thank you for any of your ideas.


title: Direction of Photo Electron Emission

text: I was looking for information on how the photo electrons are emitted when under X-ray radiation. In this ancient review paper here http://authors.library.caltech.edu/1551/1/WATpr28.pdf they state that the most common angles for non polarized X-ray beams (of various energies) range roughly at around 70-80 degrees with the beam. It is unclear to me, whether the photo electrons are moving towards the source of the X-ray beam or away from it ? Undergrad texts do not seem to shed light on this matter, the best one gets is pictures with emitted electrons being at a 90 degree angle to the incoming photon. Also, I assume this angle is given for a cone, i.e. it's 70-80 w.r.t. the beam, but with 2pi angle around the beam ?


title: Inner product of position and momentum eigenkets

text: Let's define $\hat{q},\ \hat{p}$ the positon and momentum quantum operators, $\hat{a}$ the annihilation operator and $\hat{a}_1,\ \hat{a}_2$ with its real and imaginary part, such that $$ \hat{a} = \hat{a}_1 + j \hat{a}_2$$ with $$\hat{a}_1 = \sqrt{\frac{\omega}{2 \hbar}}\hat{q},\ \hat{a}_2 = \sqrt{\frac{1}{2 \hbar \omega}}\hat{p}$$ (for a reference, see Shapiro Lectures on Quantum Optical Communication, lect.4) Define $|a_1 \rangle,\ |a_2\rangle,\ |q\rangle,\ |p\rangle$ the eigenket of the operator $\hat{a}_1,\ \hat{a}_2,\ \hat{q},\ \hat{p}$ respectively. From the lecture, I know that $$ \langle a_2|a_1\rangle = \frac{1}{\sqrt{\pi}} e^{-2j a_1 a_2}$$ but I do not understand how to obtain $$ \langle p|q\rangle = \frac{1}{\sqrt{2\pi\hbar}} e^{-\frac{j}{\hbar}qp}$$ I thought that with a variable substitution would suffice, but substituting ${a}_1 = \sqrt{\frac{\omega}{2 \hbar}}{q},\ a_2 = \sqrt{\frac{1}{2 \hbar \omega}}{p}$, I obtain $$\frac{1}{\sqrt{\pi}} e^{-\frac{j}{\hbar}qp}$$ which does not have the correct factor $\frac{1}{\sqrt{2\pi\hbar}}$. What am I missing?


title: Surely proper time expands like proper space?

text: The FRW metric at the origin $r=0$ is simply given by: $$ds^2 = -dt^2 + a^2(t)\ dr^2$$ Setting $dt=0$ gives us an element of proper distance $ds$ given by: $$ds = a(t)\ dr$$ Thus we get the well known result that space expands with the scale factor. Setting $dr=0$ gives us the relationship between an element of proper time $d\tau$ and an element of co-ordinate time $dt$ (using $d\tau^2 = - ds^2$): $$d\tau = dt$$ Thus we get the well known result that for a co-moving observer proper time is the same as co-ordinate time. However there is one more relation one can derive. We can set $ds=0$. We then obtain a relationship between an element of co-ordinate time $dt$ and an element of co-ordinate separation $dr$ : $$dt = a(t)\ dr$$ Surely this relationship implies that elements of co-ordinate time $dt$ (and thus proper time $d\tau$ for a co-moving observer) expand with the scale factor in the same way that elements of proper distance expand with the scale factor?


title: Tesla's theory of gravity

text: I was reading up on Tesla's Wikipedia page last night, and I came across this: > When he was 81, Tesla stated he had completed a "dynamic theory of gravity". > He stated that it was "worked out in all details" and that he hoped to soon > give it to the world. [75] The theory was never published. I was wondering, has this theory ever been found? Here's a link to the Wikipedia article on Tesla


title: Why can you see a light in the location that the light source just left?

text: This is my first question on this site so forgive me for the awkward wording of the question. Basically, my question is why does light from, say, a sparkler, seem to remain where it just came from to you EYE? I attached a sample photograph and i understand that cameras work differently in that they're designed to "retain" light so to speak. But why does the human eye have a similar, albeit less dramatic effect? ![alt text](http://i.stack.imgur.com/ytwcr.jpg)


title: Do two synchronous clocks have simultaneous indications?

text: Considering two clocks, $C$ and $D$, which were at rest to each other throughout a sufficiently extended trial, and given their time parametrizations $t_C : {\text{ ordered set of}}C{\text{'s indications}} \rightarrow \mathbb{R}$ and   $t_D : {\text{ ordered set of}}D{\text{'s indications}} \rightarrow \mathbb{R}$ such that Einstein's synchronism conditions were satisfied (by clocks $C$ and $D$ among each other, as well as any required transitivity condition with respect to further suitable participants/clocks), if there is an indication $C_j$ of clock $C$ and an indication $D_k$ of clock $D$ such that $t_C[ C_j ] == t_D[ D_k ]$, are such indications $C_j$ and $D_k$ called "simultaneous to each other"?


title: Electrons faster than speed of light

text: While looking at some exercises in my physics textbook, I came across the following problem which I thought was quite interesting: > It is possible for the electron beam in a television picture tube to move > across the screen at a speed faster than the speed of light. Why does this > not contradict special relativity? I suspect that it's because the television is in air, and light in air travels slower than light in a vacuum. So I suppose they're saying the the electron could travel faster in air than the speed of light in air, like what causes Cherenkov radiation?


title: Real image with converging lens?

text: I read in a AQA GCSE book that: > A real image is formed by the converging lens if the object is further away > from the principal focus/focal point. I did this experiment in class: ![Diagram](http://i.stack.imgur.com/ciRYV.png) Here is my table of results: ![Results](http://i.stack.imgur.com/BRl0x.png) However, from the quotation above, u needs to be larger than v.   * **So why in my experiment did I get that u is 15cm and v is 30cm?** Here is what I did in my experiment: A crossed wire object was placed in front of the ray box. After that, the converging lens was placed 15 cm in front of the ray box. The screen was moved until a clear, real image was visible. The distance from the lens to the screen was measured and recorded. This process was repeated with the following measurements (from the ray box to the lens): 20 cm, 25 cm, 30 cm & 50 cm. Thanks in advance


title: Transformation of angles in special relativity

text: My question is about the problem below ![enter image description here](http://i.stack.imgur.com/a5cJ5.png) Depicted are two space ships (the USS Voyager and the USS Enterprise), each with velocity $v=c/2$ relative to the space station (Babylon 5). At the exact moment the two space ships are closest together (at a distance d) the USS Enterprise fires off a shuttle with velocity $u=3c/4$ relative to the space station. The question is: Under what angle $\alpha'$ (as measured by USS Enterprise) must the shuttle be fired off in order to meet the USS Voyager? I see two possible ways to approach this:   1. Calculate the angle $\alpha$ measured by the space station Babylon 5 and figure out how angles transform when we change our frame of reference.   2. Figure out the velocity in x-/y-direction in the frame of reference of the USS Enterprise and deduce the angle $\alpha'$ from that. Now, if both ways are correct they should give the same answer. But they don't seem to, so what is wrong? * * * Here is what I have tried: Let $u_x$, $u_y$ be the velocities (in the corresponding directions) the shuttle needs to have - as observed by Babylon 5. Let $u_x'$, $u_y'$ be the velocities of the shuttle observed by USS Enterprise. Then by the rules for adding velocities: \begin{eqnarray*} u_y' &=& \frac{u_y + v}{1 + \frac{vu_y}{c^2}} \\\ u_x' &=& \frac{u_x }{1 + \frac{vu_y}{c^2}}\cdot \sqrt{1-\frac{v^2}{c^2}} \end{eqnarray*} So this gives $$\tan(\alpha') = \frac{u_y'}{u_x'} = \frac{u_y + v}{u_x} \cdot\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}$$ But on the other hand, we could imagine the guys on Babylon 5 drawing a big triangle to show the crew on USS Enterprise the trajectory the shuttle needs to take. The people on Babylon 5 will then draw a triangle with angle $\alpha$ satisfying $$\tan(\alpha) = \frac {u_y}{u_x}$$ So, if side $a$ of this triangle is parallel to the x-axis and $b$ is parallel to the y-axis, then we have $$\frac ba = \tan(\alpha) = \frac{u_y}{u_x}$$ Since for the USS Enterprise $a$ is the same, but side $b$ is contracted, they will see a traingle with side $b' = b/\gamma$ and $a' = a$. Therefore $$\tan(\alpha') = \frac{b'}{a'} = \frac ba \sqrt{1-\frac{v^2}{c^2}} = \frac{u_y}{u_x}\sqrt{1-\frac{v^2}{c^2}}$$ These are two different results in general. I cannot figure out what is wrong with either of them... Thanks for reading, help will be greatly appreciated! :)


What is the basic postulate on which QM rests. Is it that the position of a particle can only be described only in the probabilistic sense given by the state function $\psi(r)$ ? We can even go ahead and abandon the particle formalism as well. So what is the QM all about ? A probabilistic description of the physical world ? and nothing more ?


title: Why can, or can not, a perfectly incompressible fluid exist?

text: Water is normally **assumed to be an incompressible fluid** \- for example in the context of calculations involving water pressure. I wondered whether that is strictly true, or **an approximation?** Later I noticed some side note implying water is not fully incompressible. Of course that makes sense, as there are not many things "perfect" in nature in this sense, like maybe supraconductivity and suprafluidity.    Now, why, or "in what kind of way", is water only almost incompressible - is it **caused by impurities** like gases and other fluids in the water? Or is it not completely incompressible in **some fundamental way**?


title: Relation between complex dielectric constant and resistance

text: I am having a real hard time understanding **how** I reach the expression that relates resistance and $\varepsilon''$ as the complex part of the dielectric constant. I was trying to follow a clear and eloquent way of describing this. I understand how the idealized concept of a capacitor requires a $\pi/2$ phase shift between polarization and electric field which is related by the dielectric constant. If it is not $\pi/2$ , then expressing the the dielectric constant as a complex number will allow us to express that phase shift. I know that not having a phase shift equal to $\pi/2$ means that I do not have a ideal capacitor, but also a related resistance. My professors keep telling me that the following expression is a _definition_ , not a deduction: $$\varepsilon''=\sigma/{\varepsilon_0\omega} $$ i.e. $$\varepsilon''=d/{A\varepsilon_0\omega R} $$ where _d_ and _A_ are thickness and contact area of the resistance _R_ But it seems to me that I should be able to explain it better... How?


title: Is diffraction affected by interaction between photons and electrons?

text: Suppose we take a sheet of ordinary metal, make a narrow slit in it, and shine a light beam through the slit onto a screen. The light beam will diffract from the edges of the slit and spread out onto the screen. Now let's take an identical sheet, with an identical slit, but this time made of neutronium, and shine an identical light through it. Will the image on the screen be the same? Or are there additional interactions with the electrons in the first sheet which aren't present in the second experiment, and which cause the images to be different in the two experiments?


title: Is there a relation between the weak scale and the intermediated string scale?

text: I was reading these papers http://arxiv.org/abs/hep-th/0609180v2 http://arxiv.org/abs/hep-th/0610129v2 They state that $m_s$ is proportional to $M_P/\sqrt{V} $ and that $m_{3/2}$ is proportional to $M_P/V$. ($m_s$:intermediated string scale, $m_{3/2}$ gravitino mass, $M_P$:Planck mass) Then they claim that $V=10^{15} l_s^6$ is required to explain the weak/Planck hierarchy. I want to understand this statement, how is the weak scale related?


title: Why does the object $\epsilon_L Q_L + \epsilon_R Q_R$ correspond to a 16-component conserved supercharge when we have a Dp-brane?

text: I understand that when a 10-dimensional superstring theory has a Dp-brane (say, extending in the $x_0, ... , x_p$ directions) we have the total conserved supercharge given by: \begin{equation} \epsilon_L Q_L + \epsilon_R Q_R \end{equation} where $Q_L$ and $Q_R$ are the left moving and right moving worldsheet supercharges. This is because the left moving and right moving supercharges aren't conserved by string boundary conditions on the brane, while the sum of these supercharges is conserved. In the above equation we have that the killing spinors are related by: \begin{equation} \epsilon_L = \Gamma_0 \Gamma_1 ... \Gamma_p \epsilon_R \end{equation} Each $\epsilon$ is 16-component, so this last equation tells us that each component of $\epsilon_L$ is related to a component of $\epsilon_R$. As a result we only have 16 independent components of the killing spinors, and SUSY is broken in half (from 32 supecharges to 16). Now going back to $\epsilon_L Q_L + \epsilon_R Q_R$, I would expect this total conserved supercharge to be some 16 component spinor. However, don't the $\epsilon$'s and $Q$'s just contract to give some single value? How can this correspond to a 16-component spinor (total conserved supercharge) that can then be used on supermultiplets? Also, I get that $\epsilon_L Q_L + \epsilon_R Q_R$ being conserved, together with the fact that $\epsilon_L$ is directly related to $\epsilon_R$, implies that $Q_L$ and $Q_R$ aren't independent; should there not then be some equation, analogous to the second expression above, that relates $Q_L$ directly to $Q_R$?


title: Is there some sort of Pati-Salam model with mixed generations?

text: The evidence for an approximate " _lepton as fourth colour_ " symmetry is so overhelming in the particle spectrum that Hanlon's razor does not seem to apply. Still, my own incompetence fails to recognise an adequate model. Point is, once we have massive see-saw-able neutrinos, we have enough leptons to organise approximate multiplets with three colours of a quark plus one colour neutral lepton:   * $(ν_1,t_r,t_g,t_b)$ at about 174.10 GeV   * $(ν_2,b_r,b_g,b_b)$ around 3.64 GeV   * $(τ,c_r,c_g,c_b)$ around 1.698 GeV   * $(μ,s_r,s_g,s_b)$ around 121.95 MeV   * $(e,u_r,u_g,u_b)$ with null mass.    * $(ν_3,d_r,d_g,d_b)$ about 8.75 MeV But you can see the problem: there are two charged leptons with the two second generation quarks, and then two neutral leptons with the third generation! So something must be done with the L-R SU(2)xSU(2) charges of the model, or with the quark assignment. In fact I believe to remember that the paper of Harari- Haut-Weyers, from which the mass assignments for s.u.d are taken, had a model where the right and left quarks were permuted between generations, so I should expect more work to exist in the literature. My question is, do you know of some sort of "twisted Pati-Salam L-R model" where the above multiplets are valid? **EDIT 1: the representations,** To start to play, they should not be taken from the standard model, but from Left-Right symmetric models with some Pati-Salam symmetry. This means that both leptons and quarks are in representations of $SU(2)_R$ and $SU(2)_L$ with Right and Left isospins of $\pm 1/2$ where they are in the doublet and 0 where they are in the singlet. So the objection of Lubos Motl, below, is that for instance the muon above is in +1/2 of the left doublet while the strange is the -1/2 of the left doublet. And of course the same problem for the conjugate multiplet, in the $SU(2)_R$ side. But this was my original question! Is it possible to twist the symmetries _and generation-wise charge assignments_ to allow for such a mix? EDIT 1.1: To clarify, my question is on the second and third generations. The first generation multiplet $(e^L,u^L_{rgb})$ as well as its conjugate $(e^R,u^R_{rgb})$ are usual multiplets of $SU(4) \times SU(2)_L \times SU(2)_R$, the former being doublet in the L and singlet in R, the later being singlet in L and doublet in R. In this case there should be nothing surprising about a Higgs mechanism preserving SU(4), no more that the preservation of SU(3) in the Standard Model. Nobody is surprised that the three up quarks have equal mass, the three down quarks another mass but equal for all of them, and still $u^R$ and $u^L$ have different electroweak properties. EDIT 1.2: Note for instance Gabriele Honecker version of supersymmetric Pati- Salam, http://inspirehep.net/record/614377?ln=es, http://inspirehep.net/record/1185446?ln=es, where one generation has different representation that the other two. **EDIT 2: the masses (just at motivation, not the real question!)** Lubos points out that the values are numerological, but how they are? Well, this is irrelevant to the question, but it can be of marginal interest: the series is choosen so that all of the values fit with Koide formula: (174.10,3.64,1.698), (3.64,1.698,0.12195), (1.698,0.12195,0),(121.95,0,8.75). So the only input is 0 for up and 174.10 for top. Besides, the last triplet has the proportions of Harari-Haut-Wylers model: up equal to zero and $m_d/m_s$ is $\tan^2 15$. If some of you check the triplets agains Koide, remember take the negative sign for $\sqrt {m_s}$ in the second one. In this way, it is orthogonal to the charged lepton triplet. The link between the scb triplet and the charged lepton triplet is exploited to predict the masses after the breaking of the multiplets, by assuming that all the Koide equations are still happening. EDIT 2.1: when Koide is written as $m_k=M(1+\sqrt 2 \cos ({2 \pi \over 3} k + \delta))^2$, then it can be seen by inspecting above that $M_{scb}=3M_l$ and $\delta_{scb}=3\delta_l$. Asuming that this relation also survives to the breaking of "SU(4)", then it is possible to use as input the mass of electron and muon to predict all the other masses. And it works: the predictions are $173.26, 4.197, 1.77696, 1.359, 92.275, 5.32, .03564$; and the experiments (pdg2012) give, respectively, $173.5 \pm 1.0, 4.18 \pm 0.03, 1.77682 (16), 1.275 \pm 0.025, 95 \pm 5, \sim4.8, \sim2.3$


title: Terminology for opposite null lines

text: Is there a name for two null lines that lie on the opposite sides of the null cone? Each line can be obtained from the other by reflection in the axis of the null cone (the time-axis). In terms of world-lines, this corresponds to two photons moving in the opposite directions. If there is not a standard name, what would you choose to call them as a pair?


title: Why is the direction of a flame is always upward?

text: why a candle's flame direction is always upward? What is the coposition of fire? Don't say it's energy. There is something else.


title: Why is general relativity considered to be a gauge theory?

text: I have studied the first five chapters of Carroll's book (up to the Schwarzschild solution). I see similarities to the Yang-Mill theories such as the covariant derivative to account for curvature in GR which is similar to the covariant derivative to account for local gauge transformations in Yang- Mill theories. I have heard that some consider GR to be a gauge theory, but I do not understand what the redundant degrees of freedom are required to make it gauge theory.


title: Counter Rotating Bicycle wheels on same axis-> Will they still cause me to spin on a stool?

text: So people are familiar with the idea of holding a spinning bicycle wheel while on a stool (whose seat can spin). You then tilt the spinning wheel and lo and behold you start to spin on the stool. Ok fine, I get that (conservation of Angular momentum). So the real question is: if I have 2 wheels mounted on the same axis and they are spinning in opposite directions, does the angular momentum cancel out? Do you still spin if you are tilting them while on the stool?


title: Electron and photon collision

text: So electron moving left ($v_{initial}=0.8c$) collides into photon going right. After the collision electron is moving to right ($v_{after}=0.6c$) and the photon is moving to left. What is the wavelength $\lambda_{initial}$ of the photon before collision. So we are only on the x-axis and $cos\phi=-1$ because there is no scattering $\phi =180^0$. And I think we need to establish the equations for preservation of the linear momentum and energy. Something like this? $$\begin{align} E_{initial}&=E_{final} \\\ p_{initial,x} &= p_{final,x} \\\ \end{align} $$ Im not sure what are the formulas for linear momentum and energy of photon and electron. Any help? $$\begin{align} E_{initial}&=E_{final} \\\ \frac{hc}{\lambda}+\gamma m_e c^2 &= \frac{hc}{\lambda^´}+\gamma^´ m_ec^2 \\\ \end{align} $$ Where $p=\frac{h}{\lambda}$ and $\gamma =\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}$ and $\gamma^´ =\frac{1}{\sqrt{1-\frac{v^´2}{c^2}}}$ ? And for momentum:$$\begin{align} p_{initial,x} &= p_{final,x} \\\ \frac{h}{\lambda}-\gamma m_e v &= \frac{h}{\lambda^´}cos\phi-\gamma^´ m_e v^´ cos\phi \\\ \frac{h}{\lambda}-\gamma m_e v &= -\frac{h}{\lambda^´} + \gamma^´ m_e v^´ \\\ \end{align}$$ And now I have to equations with 2 unknowns $\lambda$ (what I want to know) and $\lambda^´$. Can someone verify this before I start to solve this system?:D \begin{cases} \frac{hc}{\lambda}+\gamma m_e c^2 &= \frac{hc}{\lambda^´}+\gamma^´ m_ec^2 \\\ \frac{h}{\lambda}-\gamma m_e v &= -\frac{h}{\lambda^´} + \gamma^´ m_e v^´ \\\ \end{cases} \begin{cases} \frac{hc}{\lambda}-\frac{hc}{\lambda^´} &= \gamma^´ m_ec^2 - \gamma m_e c^2 \\\ \frac{h}{\lambda}+\frac{h}{\lambda^´} &= \gamma^´ m_e v^´ +\gamma m_e v , & \text{let's multiply this with $c$} \\\ \end{cases} \begin{cases} \frac{hc}{\lambda}-\frac{hc}{\lambda^´} &= \gamma^´ m_ec^2 - \gamma m_e c^2 \\\ \frac{hc}{\lambda}+\frac{hc}{\lambda^´} &= \gamma^´ m_e v^´ c +\gamma m_e v c \\\ \end{cases} $$\lambda = \frac{2hc}{\gamma^´ m_ec^2 - \gamma m_e c^2 + \gamma^´ m_e v^´ c +\gamma m_e v c}?$$ $$\lambda = \frac{2\times1.239eV\mu m}{\frac{0.511\times 10^6\frac{eV}{c^2}}{\sqrt{1-\frac{(0.6c)^2}{c^2}}}(c^2+0.6c^2)+\frac{0.511\times 10^6\frac{eV}{c^2}}{\sqrt{1-\frac{(0.8c)^2}{c^2}}}(0.8c^2-c^2)}\approx 3.0\times 10^{-6} \mu m?$$ That's too small I think... i was expecting wavelenght in $nm$. Damn I guess it's wrong :(


title: Potential Inside Conducting Cube

text: A cubical box with sides of length L consists of six metal plates. Five sides of the box { the plates at $x=0, x=L, y=0, y=L, z=0$ - are grounded. The top of the box (at z = L) is made of a separate sheet of metal, insulated from the others, and held at a constant potential $V_0$. Find the potential inside the box. I am attempting to set up this problem with Legendre polynomials. Since the top portion is grounded, this affects the "potential" region I can take, and I am unclear on how about to go setting it up. I have set up the Laplace as follows: $1\over x$${d^2x\over dx^2} +$ $1\over y$${d^2y\over dy^2} +$$1\over z$${d^2z\over dz^2} = 0$ In which at this point, must each component equal a constant? which gives $k_x^2+k_y^2+k_z^2 = 0$ $1\over x$${d^2x\over dx^2} = k_x^2$ $1\over y$${d^2y\over dy^2} = k_y^2$ $1\over z$${d^2z\over dz^2} = k_z^2$ where z is given by: $Z(z) = sinh(k_zz)$ but at this point, how do I solve the coefficients? Or should I be using the general form and find coefficients: $x = Acos(k\theta) + Bsin(k\theta)$ $y = Ccos(m\phi) + Dsin(m\phi)$ $z = Ee^{(k^2 +m^2)^.5z}$ + $Fe^{-(k^2 +m^2)^.5z}$ I'm stuck on finding the coefficients using this method.


title: Neglecting friction on a pulley?

text: So, this is how the problem looks: http://www.aplusphysics.com/courses/honors/dynamics/images/Atwood%20Problem.png Plus, the pulley is suspended on a cord at its center and hanging from the ceiling. You're given masses of the objects, mass of the pulley and it's radius. And your assumptions are that the string is massless and inelastic and that there is **no friction** between the string and the pulley. You have to find the tension in the string suspending the pulley (it's not drawn in this particular figure). My conclusion was that, since there is no friction between the string and the pulley, the tension would have to be equal all along the string's length and the pulley would not rotate, it would slip and there would be no way to transfer the linear motion of the masses to the rotational motion of the pulley. (The resulting torque would be zero.) So, according to me, the tension in the string suspending the pulley would simply be the tension along the rope multiplied by two. But the solution includes a rotating pulley and its rotational inertia... and it gives a different answer... Where did I go wrong? And why? If not, how can I prove I'm right?


title: Feynman rules for coupled systems

text: I have the following system of two coupled real scalar fields $\sigma$ and $\phi$: $S[\phi,\sigma]=\int{d^4x[-\frac{1}{2}\partial_\mu\phi\partial^\mu\phi-\frac{1}{2}m^2\phi^2-\frac{1}{2}M^2\sigma^2-\frac{\lambda}{2}\sigma\phi^2]}$. What would the Feynman rules be for this system? I realise that something will be different about the propagator for $\sigma$ as it has no kinetic term, but I'm not sure how that translates into the rules. Furthermore, how would you draw the Feynman diagrams which determine the amplitude for a 2 -> 2 scattering process associated to the field $\phi$ (at tree level)? The whole idea is perplexing me a bit and I was wondering if there was any insight here!


title: Why do we need a Theory-Of-Everything (TOE)?

text: On the heels of some of the discussion about the recent TOE article by New Scientist, what are the key reasons why we need a TOE? The obvious immediate response is that a TOE should give us a means of making predictions in quantum gravity, but what else might be motivating the need for a TOE. Is it economical to continue the search? If this question passes muster, I request respondents avoid the typical debate surrounding string theory in the popular press. In that context, I think the appropriate question is whether a TOE should be able to solve its own ground state problem, or can one even formulate a consistent theory that can accomplish that task.


title: Is Lagrangian a scalar?

text: I may be wrong: Lagrangian are scalars. They are NOT invariant under coordinate transformations. The simplest example is when you have a gravitational potential ($V=mgz$) and you translate $z$ by $a$ (some number). $$L=\frac12m\left(\frac{dz}{dt}\right)^2-mgz\to L=\frac12m\left(\frac{dz}{dt}\right)^2-mgz-mga$$ thus the Lagrangian changed under this coordinate transformation. However Euler-Lagrange equations ARE invariant under coordinate transformations. **So some scalars do vary under coordinate transformation!** Thus components of vectors are scalars, thus time is. Again, try to say the following sentence: > the vector $a$ is the first component of the vector $b$. Can you please clarify?


title: Stripeless cleaning of windows

text: > Cross post: http://chemistry.stackexchange.com/q/4377/22 Last week I was discussing with a friend how we thought the stripeless cleaning of windows is achieved when using a cleaner like Windex Glass Cleaner as opposed to just using hot water, in which case you do get stripes left behind. At first we thought that the stripeless nature was caused by surfactants in the cleaner that allow the (polar) dirt on the windows to be collected in the water through micellation. However, when we use normal cleaner not specifically designed to clean glass/windows, we also get the stripes even though we are sure that there are also surfactants in the normal cleaner. The question is: what is the difference between the glass cleaner and the regular cleaner that makes sure that windows become stripeless clean? One thought we considered but couldn't really work out was that the stripe formation would be something caused by a combination of the marangoni effect and evaporation, similar to the coffee-stain effect which can be overcome by specific tuning of the surfactants as shown here. Could someone explain how the stripeless cleaning works? Can fluid dynamics explain what happens or is it really something chemical going on? [EDIT] When searching I also ran into a solution that (at least the manufacturers claim) will also give stripeless clean windows. The interesting thing is that they use pure $H_2O$ so without any surfactants but also without any dissolved salts like regular tap water has.


title: Time travel in the past

text: The past weeks I watched some episodes from "Through the Wormhole" with Morgan Freeman. When talking about traveling back in time, they said you can only travel back up to the date when the first time travel occured. So, if for instance time travel will be discovered in 2045, all time travelers(from any given time) won't be able to travel in time earlier than 2045. Why is this? Can anyone explain this please? I know this is just theoretical but still I would like to know the concept behind it.


title: Viscous fluid flowing around obstacle: would it deflect earlier?

text: Consider a viscous fluid, flowing linearly (say, with velocity $\vec u = [1,0]$ everywhere). Then an obstacle is put in the flow. Would a highly viscous fluid start deflecting around the obstacle earlier (more up-stream) than one with low viscosity?


title: Simple quadrupole moment

text: I have a very simple problem: There is a charge $-q$ at $(0, 0, d)$ and $(0, 0, -d)$ as well a charge $2q$ at $(0, 0, 0)$. I have to calculate the quadrupole moment using spherical coordinates. I use $x=\cos(\phi)$. The formula I build up is this: $$ q_{2,m} = \frac{(2-m)!}{(2+m)!} \frac{(-1)^{m+1}}8 \int_{-1}^1 \mathrm dx \, \left(1-x^2\right)^{m/2} \frac{\mathrm d^{2+m}}{\mathrm dx^{2+m}} \left(x^2-1\right)^2 $$ $$\times\int_0^{2\pi} \mathrm d \phi \, \mathrm e^{\mathrm im \phi} \int_0^\infty \mathrm dr \, r^2 \rho(r, x, \phi).$$ Is this even correct? Then I tried to figure out what the $\rho$ has to be, this is what I currently believe to be correct: $$ \rho(r, x, \phi) = q \delta(\phi) \left[2 \delta(r) \delta(x-1) - \delta(r-d) \delta(x-1) - \delta(r-d) \delta(x+1)\right].$$ So now I would just need to calculate the integrals, right? The problem for positive $m$ would be that $\int \mathrm dx \, \delta(x-1) \left(1-x^2\right)^{m/2}$ would be zero. And $\int \mathrm dr \, r^2 \delta(r)$ would also be zero. So the $q_{2,1}$ and $q_{2,2}$ are zero. I previously used the following charge density (although there was a negative radius): $$ \rho(r, x, \phi) = q \delta(x) \delta(\phi) \left[2\delta(r) - \delta(d-r) - \delta(d+r)\right].$$ And I got: $$ q_{2,-2} = 3! q d^2 ,\quad q_{2,-1} = 0 ,\quad q_{2,0} = -q d^2 ,\quad q_{2,1} = 0 ,\quad q_{2,2} = \frac 14 qd^2. $$ In one of the books, it says that only the $Q_{3,3}$ element of the tensor would be non-zero. So how can I obtain the correct answer here?


title: Gas kinetic representation of trans-critical conditions

text: From a molecular point of view, can we think of the super-critical conditions as conditions where T and p are large enough that the collisions of gas molecules are frequent and powerful enough to prevent the molecular bonds that would form a liquid cannot be sustained. So if we consider a single fluid in a constant pressure system and with a temperature gradient from sub-critical temperatures to super-critical temperatures, how accurate is this diagram? ![Trans-critical event from a molecular dynamics point of view](http://i.stack.imgur.com/NgSSC.png) How can we represent the disappearance of surface tension on the right side? Are molecules being stripped from the surface by collisions? Because the molecules inside the compressed liquid are still bonded right? What would you change in this crude diagram to make it more representative of what really happens? To give a bit more context, I'm more of a fluid mechanics guy modeling inject. So I might not be using the exact terms that I should. The terminology I use is within the context of this diagram: ![O2 phase diagram](http://i.stack.imgur.com/W1wRj.png) Phase diagram of oxygen in the $p-T$ space. The black dot represents the critical point at the intersection of the long-dash lines for the critical temperature and pressure. The short-dash line is the pseudo-boiling line (Oschwald_2006a) or Widom line (Simeoni_2010a) and corresponds approximately to the region of maximum specific heat at constant pressure. The density is represented by the colored contours in the background.


title: Logical connection of Newton's Third Law to the first two

text: The first law and second laws of motion are obviously connected. But it seems to me that the third law is not related to the first two, at least logically. (In Kleppner's Mechanics the author states that the third law is a necessity to make sense of the second law. It didn't make sense to me, though. I'll post the excerpt if anyone would like to see it.) Thanks, Ron EDIT: Excerpt from Introduction to Mechanics by Kleppner & Kolenkow (1973), p. 60: > Suppose that an isolated body starts to accelerate in defiance of Newton's > second law. What prevents us from explaining away the difficulty by > attributing the acceleration to carelessness in isolating the system? If > this option is open to us, Newton's second law becomes meaningless. We need > an independent way of telling whether or not there is a physical interaction > on a system. Newton's third law provides such a test. If the acceleration of > a body is the result of an outside force, then somewhere in the universe > there must be an equal and opposite force acting on another body. If we find > such a force, the dilemma is resolved; the body was not completely isolated. > [...] > > Thus Newton's third law is not only a vitally important dynamical tool, but > it is also an important logical element in making sense of the first two > laws.


title: Does performing a measurement on a system change its internal energy?

text: I'm studying Quantum Mechanics in my spare time from a general point of view (no technical details) so some fundamental question came into my mind: How is it possible to detect a single photon without making any change to it? How does observation is related to the total energy of a system and does the act of performing measurement alter the internal energy of the observed system as a general rule?


title: How does a state in quantum mechanics evolve?

text: I have a question about the time evolution of a state in quantum mechanics. The time-dependent Schrodinger equation is given as $$ i\hbar\frac{d}{dt}|\psi(t)\rangle = H|\psi(t)\rangle $$ I am self-learning the quantum mechanics so there are few conceptual questions confused me. I am reading a book, in which the author emphasizes that any state could be expanded in terms of the eigenstates and eigenvalues of Hamiltonian. So does it mean in above equation, it is not requiring $\psi(t)$ to be an eigenstate (i.e. it could be anything)? I am wondering it is only true when we are trying to expanding a steady state in terms of the eigenstates, that is $$ c_n(0) = \langle E_n|\psi(0)\rangle $$ But if the state is time-dependent, the expanding coefficient $c_n$ is not longer stationary but evolving in time, the book gives $$ c_n(t) = c_n(0)\exp(-iE_nt) $$ where $E_n$ is the eigenvalue. So the time evolution of the state is $$ |\psi(t)\rangle = \sum_n c_n(t)|E_n\rangle $$ is it correct? I understand the math. But physically, how do we understand the initial coefficient $c_n(0)$ evolves in the exponential form? and why the exponential has eigenvalue $E_n$? It looks to me that different eigenvalue plays different role in time evolution, why is that? If we look into the math, seems that the contribution of the eigenstate with higher 'energy' (eigenvalue) is decreasing faster in time, what's the physical reason for that? The last question is even confusing. The text said the system could has many different states. But when you measure it, only one of the eigenstate will be shown and the measured value will be the corresponding eigenvalue. I don't understand this statement but just assume it is true. So if the state of the system is initially $|\psi(0)\rangle$ and it evolves in time as $|\psi(t)\rangle$. I measure the system at time $t=\tau$, so at this time, is it still true that the outcome must be one of the eigenvalues even the state is time-dependent?


title: Observations in the cathode ray tube experiement

text: ![enter image description here](http://i.stack.imgur.com/6drQ1.png) 1.One of the observations I learned was that the glass tube begins to glow with a brilliant green light. Many websites I read through refer to a fluorescent material. However, as shown in the above diagram there was no fluorescent material in the experiment carried out first on the cathode ray tube. **So where does the green glow come from. Is this the color of the radiation itself?** 2." _Cathode rays travel in straight lines. That is why, cathode rays cast shadow of any solid object placed in their path. The path cathode rays travel is not affected by the position of the anode._ " I just can't seem to understand this explanation of the one of the observations.Also, different websites analyses this observation differently. For example, " _The cathode rays consist of material particles because they produced shadow of objects placed in the way_ " 3.Two of the conditions of the experiment were air at very low pressure and secondly a very high potential difference. Could someone please tell me why these conditions were necessary? I know the questions are very silly but because different websites refer to different things, I am becoming confused with something that should be simple to understand.


title: Stellar Activity Cycle versus Metallicity

text: Our Sun exhibits sunspot max/mins on about a 11 year period. It's a G2 spectral class on the HR Diagram. We know that some stars exhibit "starspot" cycles of various intensities and periods. These cycles are the result of changing magnetic activity within rotating stars. Would a star identical to our Sun except metallicity, have the same cyclical behavior? Metals (elements heavier than helium), comprise about 2% of the Sun's mass. A metallicity increase for example, would imply a smaller radius for equal mass. Wouldn't a Sun-like star of higher/lower metallicity rotate faster/slower, which would speed up/ slow down its activity cycle?


title: Dust generated static space-time implications on fluid 4-velocity

text: Imagine we have a perfect fluid with zero pressure (dust), which generates a solution to Einstein's equations. Show that the metric can be static **only if** the fluid four-velocity is parallel to the time-like (and hypersurface orthogonal) Killing vector characterizing the static metric. So, I know the following things: Energy momentum tensor of dust with 4-velocity field $u^{a}$: $T_{ab} = \rho u_{a}u_{b}$ Hypersurface orthogonality condition of time-like killing vector $\xi ^{a}$: $\xi_{[a}\nabla_{b}\xi_{c]} = 0$ I haven't been able to come up with much. We know that $\nabla_{a}T^{ab} = 0$ and $u^{a}u_{a} = -1$. This tells us that $\nabla_{a}(\rho u^{a}u^{b}) = \rho u^{a}\nabla_{a}u^{b} + u^{b}\nabla_{a}(\rho u^{a}) = 0$ and $u_{a}\nabla_{b}u^{a} = 0$. Hence, $\rho u^{a}u_{b}\nabla_{a}u^{b} + u_{b}u^{b}\nabla_{a}(\rho u^{a}) = 0 \Rightarrow \nabla_{a}(\rho u^{a}) = 0$. This further implies that $\rho u^{a}\nabla_{a}u^{b} = 0$ [B]identically[/B] thus $u^{a}\nabla_{a}u^{b} = 0$ so the dust travel on geodesics. However I am having a hard time figuring out how any of this will be useful (if at all) in connecting it to $\xi ^{a}$ and the fact that $\xi_{[a}\nabla_{b}\xi_{c]} = 0$. I highly doubt any of the above will be useful though, it doesn't seem like it would be. So the goal is to show that if $\xi^{a}$ is in fact a time-like and hypersurface orthogonal killing vector field then $u^{a} = \alpha \xi^{a}$ where $\alpha$ is just the normalization factor. It seems one possible route would be as follows: if I am in the coordinate system $(t,x^{1},x^{2},x^{3})$ adapted to the time-like KVF, i.e. the one where the time derivatives of the metric vanish and $\xi ^{a} = (\frac{\partial }{\partial t})^{a}$ (which in the coordinate basis is just $\xi ^{\mu} = \delta^{\mu}_{t}$), then this amounts to showing that $u^{i} = 0$ (where the $i$'s run over the spatial indices) because then $u^{\mu} = \alpha \delta ^{\mu}_{t} = \alpha \xi ^{\mu}$ and this expression is covariant so it would then have to be true for all coordinate systems which is what we want. Apparently this has to somehow use the fact that $u^{a}$ is the 4-velocity field of dust. However I am unsure of how to go about showing that $u^{i} = 0$ or even if this approach is practical / doable.


title: Resistors in series vs parallel

text: I understand that for resistors in series we need to add R for each resistor to get the equivalent resistance, where R is resistance. I also understand the mathematical manipulations that show that for resistors in parallel we need to add the 1/R for each resistor to get the equivalent resistance. However I have a couple of questions that have to do with the physical interpretation of the mathematical manipulations involved. First, I am trying to understand why in the case of parallel resistors we have to add 1/R's instead of adding R's as in the case of resistors in series i.e. could it be thought that the current in the wire is like a sort of water flow in the pipe that has two horizontal parallel pipes positioned one on top of the other and in between connected by a number of vertical pipes that have a tap each that could be adjusted to represent different resistances. Then at the top of each vertical pipe the water pressure P exists and at the bottom the pressure will be reduced by P*R. For example, suppose we have three vertical pipes, so then the pressure at the bottom of each (after the tap) is P*R for the total of 3*P*R so why is the equivalent resistance not 3*R*P but 1/R = (1/R1 + 1/R2 + 1/R3) and how can I be able to visualize this relation?


title: Where does matter come from?

text: I admit, it's been a few years since I've studied physics, but the following question came to me when I was listening to a talk by Lawrence Krauss. Is there any knowledge of from where matter that exists today originated? I recall that the law of conservation of mass asserts that matter cannot be created nor destroyed, but surely the matter we see today had to be created at some point? Perhaps I am applying this law in the wrong fashion. The reason I ask, is because Krauss mentioned that the elements of organic matter where created in stars, not at the beginning of time (whenever that may have been), but I ask, where did the building blocks for these elements arise? Were they too created in stars? If so, from where did their constituent building blocks come? Please forgive me if this off topic, it is my first post on this particular stackexchange site. Thank you.


title: Characteristic waves in a plasma

text: What does "characteristic wave" mean in the context of plasma waves? For example, when propagating parallel to the plasma magnetic field, the characteristic wave is circularly polarized. Does this mean that only a circularly polarized wave can propagate in this direction? Could an elliptically polarized wave travel in this direction instead?


title: Whats the range of dimension exponents in the SI Units system?

text: I am working on a software library for Units of Measure. To represent dimensions, I need to know the required range of exponents for each of the seven base units (precisely, I need to know the required maximum number of bits for each exponent). After reading many related questions, I still have some questions. As far as I understand, all physical quantities can be expressed as the product of the seven base units, with each base unit raised to the power of 0, +-1, +-2, +-3, and so on. So the dimension of a quantity is represented by a vector / array of 7 integers. E.g. for volume (length ^ 3) that array would be (3, 0, 0, 0, 0, 0, 0), for acceleration (length / (time ^ 2)) it would be (1, -2, 0, 0, 0, 0, 0). While I could calculate the square of a volume (with dimension 6,0,0,..), it doesn't make much sense, because I have never encountered any physical quantity that needs (length ^ 6). My questions: a) What is the maximum required exponent for Length ? Do you know any example where +-3 is not enough ? b) Maximum exponent for Time ? Any example beyond +-4 ?   _(update: raised to 4 for capacitance)_ c) Maximum exponent for Mass ? Any example beyond +-1 ? d) Maximum exponent for Electric Current ? Any example beyond +-1 ? e) Maximum exponent for Temperature ? Any example beyond +-4 ?   _(update: raised to 4 for Stefan-Boltzmann constant)_ f) Maximum exponent for Amount of Substance ? Any example beyond +-1 ? g) Maximum exponent for Luminous Intensity ? Any example beyond +-1 ? .. plus three bonus questions: h) Is there some required order of the seven base units ? I have seen more than one .. i) Are those seven base units enough for everything, or is there some bleeding edge science that needs more ? j) Are the exponents always integral ? Never ever something like ^ 1.5 or ^ e ? Thanks. (update)   I guess I should clarify:   I am aware that **values** might need rational or real exponents.   I am aware that **intermediate** results might need a wider range of integer exponents.   I have read the relevant SI standard document, and lots of websites about the topic. My questions are just about the base units exponents that define the dimension of (base or derived) units. The most precise statement i found so far was: _the exponents are very small integers_. However, to exploit all filthy speed tricks of CPU or GPU assembly programming, I need to know the exact number of bits required.


title: Critical dimension in quantization of p-branes

text: So I have what might be a fairly basic question, but my understanding that in the quantization of the the string, or the 1-brane, there are conditions on the number of spacetime dimensions to ensure that the Lorentz algebra still holds. For the bosonic string we have D=26 and for the superstring we have D=10. Now I know that an equivalent condition on the critical dimension comes up by requiring that the conformal anomaly vanish. This is the prelude to my question: Are there any similar or equivalent restrictions on the number of spacetime dimensions are come up in the quantizing more general branes? Is it only the string because only the worldsheet theory has conformal invariance that we want to ensure holds in the quantized theory? If this statement is equivalent to the Lorentz algebra closing, why for a general p-brane do we not need to impose any conditions on the dimensions for this to hold?


title: localized electrons in the crystals

text: Why electrons in low lying levels of individual atoms stay localized in their own atoms in a crystal? Doesn't this contradict Bloch's theorem?


title: Center of mass error - calculating systematic error in change in PE

text: Suppose we have to calculate systematic error in change in PE. Let's suppose systematic error due to scale is 1%. I'm confused about the center of mass error. \begin{align} \Delta PE = m*g*h_1 - m*g*h_2\\\ \end{align} However, expression for calculating systematic error is \begin{align} \sigma_{\Delta PE} = \sqrt{(m*g*\sigma_{h_1})^2 + (m*g*\sigma_{h_2})^2}\\\ \end{align} Does any error due to center of mass get canceled out because unlike scale, it's not a percentage, but a fixed amount? Or does it get incorporated somehow? If it gets incorporated, how? I've come up with something like below, but I don't think it's right: \begin{align} \sigma_{h_1} = \sqrt{(\sigma_{h_{scale}})^2+(\sigma_{h_{CofM}})^2} \end{align}


title: what is uniform velocity?

text: i have a very basic question from school days. what does it mean to say an object is moving with uniform speed? it seems to me now that it should be an unit dependent concept. for example if speed is the derivative of distance traveled, i.e. $X'(t)$ , and I decide to measure distance on a new scale $F(X)$, a monotonic function of $X$, but not a linear multiple. Then, speed in that scale at a point t would be $F'(X(t))$. $X'(t)$ , which will not be constant as $F$ can be an arbitrary increasing function. and if indeed "uniformness of speed" is a unit dependent concept, what does it mean to say, light travels at a constant speed ? also this would mean "uniform acceleration" is also a unit dependent concept. how is then, "acceleration due to gravity" a universal constant ? would it cease to be a constant if I measured acceleration in $\log(m/s^2)$ ?


title: Doppler effect from an accelerating frame of reference

text: The standard equation for the doppler effect from a frame of reference with constant velocity is $f_0$ initial frequency; $v$ is the velocity of waves in the medium; $v_r$ is the velocity of the receiver relative to the medium; $v_s$ is the velocity of the source relative to the medium; $$f = \left(\frac{v + v_r}{v + v_s}\right)f_0$$ Now I'd like to know what's the equivalent for an accelerating frame, I have $$f = \left(\frac{v'\text{dt} + v_r}{v'\text{dt} + v_s}\right)f_0$$ But it just doesn't seem correct. EDIT Mea Culpa - I was thinking of the signal being emitted from an accelerating train B as being the frame of reference. So, second attempt. 1)You are standing at a platform. 2)The wind is stationary. 3)A train is accelerating towards you - blowing its whistle. What is the formula for the frequency received at the station? Where the standard formula is $$f = \left(\frac{v }{v + v_s}\right)f_0$$ rewriting for an accelerating frame of reference for the source $$f = \left(\frac{v }{v + (dv_s/dt)dt}\right)f_0$$ but I don't think this goes anywhere, and yes a non-linear solution is probably the answer but it's like a three legged black dog with rabies - it's better than no dog at all.


title: Finding the spectrum of a polynomial of the creation and annihilation operators

text: Is there a general algorithm to find the spectrum of $S S^\dagger$, where $S$ is a homogenous polynomial (of degree $n$) of the annihilation operators (of $d$ variables)?


title: Why should the Fermi level of a n-doped semiconductor be below the one of a p-doped?

text: In a pn-junction, the difference in Fermi level between the p doped and the n doped regions causes the apparition of a built-in electric field at equilibrium. This electric field goes from the n to the p, (so the positive carriers, for example, would not feel anymore the Coulomb attraction from the ionized atom donors), meaning that the Fermi level of the n doped region is below the one of the p doped, but I don't see any elementary argument explaining that.


title: Relativistic basic question - four vector, Lorentz matrix

text: I have heard relativistics only very compressed during my student time. Now I looked up the definitions again and a question comes into my mind: A contravariant vector is transformed like this: $(a^\mu)'=L_{\mu\lambda}a^\lambda$ wherein $L_{\mu\lambda}$ is the Lorentz matrix. A covariant vector is transformed like this: $(a_\mu)'=L^{-1}_{\lambda\mu}a_\lambda=L^{T}_{\lambda\mu}a_\lambda=L_{\mu\lambda}a_\lambda$. Does this mean no matter if I have a co- or contravariant four vector I use the same matrix to transform them from one coordinate system to the other? And one general question: Why do I need co- AND contravariant four vectors? Isn't one enough? **EDIT: My definitions** I think I found the solution. We do not have the common definitions of the four vectors! I have a definition like this: $a^\mu=(a_1,a_2,a_3,\mathrm{i}ct)$ The normal definition is: $a^\mu=(a_1,a_2,a_3,ct)$ Then, regarding the answers, it doesn't matter where I put the indices of the Lorentz matrix, does it? Within the new definition, the Lorentz matrix becomes non-symmetric $L=\left(\begin{array}{ccc}\cosh \theta &0&0&\mathrm{i}\sinh \theta \\\ 0&1&0&0\\\0&0&1&0\\\ -\mathrm{i}\sinh&0&0&\cosh \theta\end{array}\right)$ In contrast, the normal Lorentz matrix is $L=\left(\begin{array}{ccc}\cosh \theta &0&0&\sinh \theta \\\ 0&1&0&0\\\0&0&1&0\\\ \sinh&0&0&\cosh \theta\end{array}\right)$ Now we see that indeed $(a_\mu)'=L^{-1}_{\lambda\mu}a_\lambda=L^{T}_{\lambda\mu}a_\lambda=L_{\mu\lambda}a_\lambda$. So in this representation, the transformation of co- and contravariant four vectors are done by the same matrix. Can anyone confirm?


title: Energy non-conservation for time-dependent potentials

text: Written in a book I read that the "total energy is not preserved when the potential depends explicitly on time", i.e. $U=U(x,t)$. Is there any proof or explanation for this?


title: What is approximately critical Q for the space shuttle?

text: Everything is in the title, I wonder what was approximately the critical dynamic pressure the space shuttle (RIP) could handle, as I am trying to create a simple space flight simulator. I don't need a very precise value, there will only be futuristic space ships but I don't want to be too far from reality.


title: Propagating degrees of freedom of graviton

text: What is the best way to see that the number of propagating degrees of freedom or gravitons in 3 dimensions is $0$ ? By graviton I mean the metric and _NOT_ some topologically massive graviton that one can include in the Lagrangian. Also is the number in 4-dimensions 2?


title: Question on Radiance equation

text: The radiance equation is $$ L = \frac{d}{dA} \frac{2(\phi)}{dW cos(\theta)} (watt/srm^2) $$ where $\phi$ is the flux. I am thinking, should not be the cosine term on the numerator instead of the denominator? Having the cosine in the denominator will make L goes to infinity if $\theta = 90$ , which does not make sense to me. My understanding is that if $\theta = 90$ (i.e. flux direction is perpendicular to the surface normal), then $L$ should equal to zero (and not infinity).


title: Is it possible that universe might not be speeding up expansion?

text: I'm not sure but I was thinking of galaxies shrinking with time while still moving apart from each other at almost a constant speed or less (i.e: uniform/slightly decelerating expansion). This may need further evidence such as tracking the distance between stars inside our galaxy to check if they are getting closer. If so, would this make an alternative explanation for visualizing the universe as if it were expanding in an accelerating rate?


title: What causes the collapse of a Magnetic Field?

text: It's theorized that Mars once had a magnetic field which collapsed at some point. There's also lots of speculation that a similar process will eventually happen to Earth. More recently has been the information that the earth's magnetic field is slowly dropping, presumably preparing for its 'flip' that happens every few hundred thousand years. But what causes a magnetic field to eventually collapse to the point that it ceases to be?


title: If the moon was rapid enough would it be able to orbit the earth from a close distance?

text: If the moon was close in orbit that it's surface was like 100 km away from the earth's surface. And it had a large enough angular velocity will it be able to hold orbit? If this was possible, is something similar possible to exist in the universe (the very close orbit)? If this was hypothetically in effect (ignoring the damage that would cause to life on earth), would a person on earth, or a person on the moon tend to levitate, or gain weight on the far sides?


title: Increase in number of micro states explanation or restatement of second law?

text: Is the boltzmann's expression of entropy as log of micro states leading to the formulation that system is more likely to be in a macrostate with more no. Of micro states really is an explanation or restatement of second law of thermodynamics ?


title: What is the physical meaning of charges at light-like infinity in asymptotically flat space-times?

text: In the case of charges defined at space-like infinity, I can understand the physical meaning of them because they can be related to measurements made by a physical observer (that is an observer whose wordline is time-like). For example in four dimensions, for the Schwarschild solution, the ADM mass coincide with the mass measured by an observer that is motionless (in Schwarschild coordinates) and far from the center of the solution. My puzzle is then the following : since no physical observer can reach light- like infinity, how are physically interpreted the charges defined at light- like infinity? How are they related to any measurement? Many thanks for your help.


title: Quantum physics and object location

text: If I understand correctly, according to quantum physics, subatomic particles can "leap" from one location to another (for example, from a location on earth to a location on the "other side" of the universe). Is this property unique to subatomic particles or is it possible (no matter how highly improbable) for objects of any size to do that?


title: Statistical Mechanic

text: One can define entropy as $$S=k\log{\omega(E)},$$ where $\omega(E)$ is the numbers of states with energy equal $E$; and the canonical partition function for a set of N particles is defined as$$Z_N=\sum_{\phi}e^{-\beta E[\phi]}=e^{-\beta F(\beta,N)},$$ where the sum run on states $\phi$ and the free energy is defined as $F(\beta,N)=U-TS.$ The mean value of the internal energy and the entropy i learned that would be $$\langle U\rangle=\frac{\partial(\beta F)}{\partial\beta}$$ $$\langle S\rangle=\beta^2\frac{\partial(F)}{\partial\beta}.$$ For definition, the mean value of any physical observable is $$\langle O\rangle=Z_N^{-1}\sum_{\phi}O[\phi]e^{-\beta E[\phi]}.$$ I'm quite sure that there will be a problem of amount $k$, if one verify the definition of $\langle S\rangle$. Am i wrong?


title: Circular Polarization

text: As one observes an clockwise (cw) circular polarized electromagnetic wave which is reflected off a denser medium or metal interface it is changed to a counter-clockwise (CCW) polarized em wave plus a phase of PI. A counter clockwise polarized wave which is reflected off a lower dense medium interface only get an additional phase of PI.   1. This means the reflected wave always has the opposite polarization?   2. But **ONLY** because the reference coordinate system is changed (since by definition of polarization the wave always travels away from the observer) and NOT because of some phase which is added only to one or the other linear polarization axis?


title: Electric Flux Contradiction

text: I am currently reading about electric flux; and from this passage I am reading, I am sensing a bit of a contradiction: "If the E-field is not perpendicular to the surface area, then the flux will be less than EA because less electric field lines will penetrate A. Consider the wedge shape surface below. The electric field lines are perpendicular to the surface area A' but not to A. ![enter image description here](http://i.stack.imgur.com/UHr1y.jpg) Since the same number of electric field lines cross both surfaces, the flux must be the same through both surfaces." Clearly, the surface A is not perpendicular to the electric field, but surface A' is. So, the number of electric fields lines passing through A should be less than the number passing through A', as they suggest in the passage before the picture. Yet, they go on to say that the number of electric fields lines passing though each surface is the same. What is going on?


title: In Noether's theorem, what is a "classical solution of the equations of motion"?

text: I'm reading a book which states that: > for each generator of a global symmetry transformation, there is a current > $j^{\mu}_{a}$ which, _when evaluated on a classical solution of the > equations of motion $\phi_{cl}$_ , is conserved. I.e. > $\partial_{\mu}j^{\mu}_{a}\vert_{\phi=\phi_{cl}} =0 $ I get the general principle, but I'm uncertain about the bit I have made _italics_. Can anyone shed some light on this?


title: High Young's Modulus and Tensile Strength of Carbon Nanotubes

text: I was recently reading about Carbon Nanotubes having extremely high Young's moduli, as well as high Tensile Strength, making them very interesting fibers. However, when I read this I wondered what was behind these great properties. After doing some searching, I found that it has to do with the sp2 bonds the carbon atoms make, which are the (one of the?) strongest chemical bonds atoms can make. Moreover, a paper mentioned that the high degree of curvature might also play a role here. But how exactly do these strong bonds and high curvature lead to a high Young's modulus, and a high tensile strength? What does curvature have to do with either of these two? I would be very happy if someone could give me a somewhat qualitative argument, as to why these fibres have these properties. If it is possible to point me towards the theory section of a paper that discusses this, that is also perfectly fine.


title: Amplitude of Probability amplitude. Which one is it?

text: QM begins with a Born's rule which states that probability $P$ is equal to a modulus square of probability amplitude $\psi$: $$P = \left|\psi\right|^2.$$ If I write down a wave function like this $\psi = \psi_0 e^{i(kx - \omega t)}$, I find $\psi_0$ inside. If $\psi$ is called **probability amplitude** then what is $\psi_0$ called? Is it perhaps called an **amplitude of a probability amplitude**?


title: Lifetime of a horizontal branch star

text: I'm trying to determine what the lifetime of a star on the horizontal branch of the Hertzsprung-Russell diagram would be. Some websites say $10^8$ while others said $10^7,$ so I don't know which one is correct or incorrect. I know it must be shorter than that of the main sequence (which is about $10^{10}$ years) but by how much and how do you know? Do you use the stellar equations?


title: What does it mean for a Hamiltonian to be SU(2) invariant?

text: Can somebody explain what it means when one says a Hamiltonian is SU(2) invariant? I know Heisenberg Hamiltonian is SU(2) invariant but why?


title: Rabi oscillations and two level dynamics

text: I'm currently looking at Rabi Oscillations, and not I have a look at the following equations: $$W = \sqrt{\Omega^2+\delta^2}.$$ The amplitude: $$\frac{|\Omega|^{2}}{\delta^{2}+|\Omega|^{2}}$$ Now, I see that if the detuning $\delta = (\omega - \omega_{0})$ is large, the two level approximation is bad, since the ratio between level 2 and 3 is getting closer to 1. But why does the two level system not work for high $\Omega$?


title: What is the distinction between thermal and electrical conductivities?

text: Aren't they both caused by electron motion?


title: How does concave bowl amplifies the sound?

text: I recently came across an article / answer in Quora see here that mentioned that if we place the iPhone in a bowl and play music, the volume is amplified. What is the physics principle behind this?


title: Why are there no explicit calculations in some articles

text: I am always baffled how in all the articles (theoretical ones), there is no explicit calculations, or at least some reference how to find it, or where to find it (the calculation). I get that, for example, in articles about quantum gravity, one would have 30 or 40+ pages paper if you'd include a whole calculation, but it's so frustrating for me, as an undergraduate student , to read an article and see that something is either "trivial", or just given the result without at least some steps given. How am I supposed to learn something, if I cannot find reference on how it's done? And most of the textbooks are giving you just the bare minimum to understand it. Or the reference on how it's done is another article where everything seems like it fell from the sky. I'd like to see how someone came to some solution explicitly. I often try to derive the result the author gave, and I get stuck at one point and I have no idea how he got the result, and that gets pretty frustrating. I am lucky I have physics SE so I can ask here, and someone who has a much greater knowledge is glad to help, but if P.SE (or other SE sites) didn't exist, I don't know were I'd find all the explanations :\ How does one learn from all this? : (


title: Accelerated charge in relativistic frame

text: An accelerated charge radiates energy as electromagnetic waves. Imagine a point charge at rest in an inertial frame We are standing on a frame accelerating with respect to the charge. Now with respect to our frame, the charge is accelerating in the opposite direction. i.e actually the force is doing work on our frame to increase our kinetic energy. _Will there be any electromagnetic radiations from the charge? Will it be observable if there is any and **where did the charge acquire that energy?_**


title: Why is $c$ considered as the speed of the photons?

text: Maxwell equations brought $\ c_{o}=\frac{1}{\sqrt{\mu_{o}\varepsilon_{o}}}\ $. Since this is a constant, it made all physicists at that time wonder where was the frame of reference? They ended up with an _aether_ answer until the Michelson–Morley experiment denied it. Now there was no frame of reference, but the genius of Einstein resolved it. However, although Einstein's theory is mathematically good, it does not imply there can not be other ones mathematically good. I just read an article written by someone not knowledgeable about physics who had this interesting idea: There is no need to consider c as the speed of the photons unless natural evidence tell us that, we could consider c as the difference between the speed of the photon that just appeared (c value) and the one at (0 value) that is going to speed up and the result is a constant speed as well, c, for all references. That is what he says, if I understood it correctly. Here is the article: http://jesusgonzalezllorente.com/ http://vixra.org/abs/1202.0013 The pictures are horrible though. So, is it actually possible to consider light as he states or is there any logical, mathematical physical law that does not allow it?


title: Symmetries of separable potential

text: For separable potential, say $x^4+y^4$, its symmetry are degenerate. Is that a generic case to every separable potential? I will explain my question: The potential $x^4+y^4$ has $A_1, B_1, A_2, B_2, E$ symmetries. However, all of them are degenerate. Quantum mechanically, the energy level are all degenerate and one can consider, for example, only the $A_1$ symmetry, why?


title: Is there any possibility in the future that domestic power consumption could be wholly solar powered?

text: Assuming a standard sized, mid terrace house with four bedrooms, a kitchen with fridge, electrical cooker and washing machine, a bathroom with shower. What would be normal daily energy consumption over the year? Could this feasibly be extracted by all the sunlight that falls on the building, that is on the roof, and building walls, assuming 100% efficiency? (My future science-fictiony solution is a coat of paint on the building which has the property of photo-synthesising at 100%! (But of course much likely a lot less)) I just want to see just how feasible this would be without going into how one could manufacture such a paint.


title: Defining left and right independent of a human body?

text: Is it possible to define right and left independent of the asymmetric human body? I am unable to think of such a definition without circular reasoning. Example:   If you are facing east, your left hand side is the one pointing to North. (But did we not define North as being on the left hand side while facing East?) Another example: Your heart is on the left side or is the left side defined by the position of heart.


title: Einstein's Famous Thought Experiment Contradiction

text: Putting Special Relativity into the General Relativity category as is current practices submerges important aspects of Einstein's 1905 paper, which I recently read in a 1952 Dover paperback (The Principle of Relativity). That paper is totally unlike the modern presentations in texts. I noticed a marked discrepancy of the analysis in that first paper with what was actually described as happening. This has to do with the famous thought experiment. I modified that experiment and still found a discrepancy that I would like explained. For the thought experiment, Einstein calculates a round trip time for a pulse of light from inception to reflection back to the inception point as measured in a moving frame. The round trip time is given as $T = (2D/c)/\sqrt {1 - {{(v/c)}^2}} ,$ where D is the distance from the inception point to a mirror moving in the same frame. Einstein’s hypothesis is that the on-board time is dilated to the same extent as the length is contracted, such that the on-board RT time as measured is independent of whether the frame is moving or not, hence, giving rise to the notion that a moving frame cannot know it is moving without an outside reference. To the contrary, any symmetrical effect, such as the dilation and contraction, no matter what their forms, cancel in the above equation. This leaves an RT time discrepancy that is clearly measurable by the on board system, which knows what the speed of light is and what the distance D is. This then contradicts the hypothesis that the moving observer cannot know or measure that they are moving. Furthermore, they can tell how fast they are moving. Let us modify the thought experiment. I put a clock at both the inception and the reflection points, both synchronized, and perform the same experiment. I measure when the pulse is reflected. All dilation and contraction effects are the same for all on-board systems and paths. I launch the pulse, measure the outbound transit time and the reflected transit time. I then compare the results. The two times are asymmetrical. Their combined time is the total time calculated above. Hence, the on-board system knows how fast it is moving. If one further modifies the thought experiment into a radar experiment with external retro-reflectors on an aircraft, one finds the same asymmetries and the same time differences comparing an approaching aircraft with a receding aircraft. In this case, only the Lorentz contraction would be occurring, but the asymmetry is still present. How is it possible to reconcile the above conclusions with the modern notions that the moving platform cannot know it is moving? I'd prefer an answer using the same math and logic Einstein used in his first paper.


title: Pushing, inertia of a cart system

text: I have a cart with another cart on top which gets pulled down by another cart on the side. There is no friction. The question is: > How strongly do I have to push with $F$ to keep the cart $m_1$ stable? ![](http://wstaw.org/m/2012/02/04/m62.png) I work in the system of the cart $m_3$. This system accelerates with some $a$. The force of inertia pulls back on $m_1$ with $m_1 a$. That is opposed by the gravitational force of $m_2$, which is simply $m_2 g$. The acceleration that I need for this to be stable is: $$ a = \frac{m_2}{m_1} g $$ Here comes the point I am not too sure about: The force is on $m_3$ and on $m_2$, so the driving force would be this: $$ F = (m_2 + m_3) \frac{m_2}{m_1} g $$ On second thought, I think also need to accelerate $m_1$ some way or another, so that my total force would be a little higher: $$ F = (m_1 + m_2 + m_3) \frac{m_2}{m_1} g $$ I prefer the latter, but what is the right solution?


title: Does optical fiber preserve the angles of incoming light?

text: I am reading up on optical fibers and there's one thing I haven't understood yet:   * Does an optical fiber preserve the angles of incoming light?  For example, if we light two LEDs in front of an optical fiber (within its acceptance cone) and hold the other end of the fiber in front of a paper, would we see two separate light dots on the paper (because the angles of the incoming light were preserved), or just one (because the angles were messed up...)? (Another way of posing this question: What would I see if I looked into the end of an optical fiber – would I see what's in front (within the acceptance cone) of the other end of the fiber or just a blurry dot? - disregarding I would need extremely good eyesight etc...)


title: Which direction will the yoyo move?

text: This question has been around the net for a while, and I haven't seen a good explanation for it: ![enter image description here](http://i.stack.imgur.com/XezSb.gif) A yo-yo is initially at rest on a horizontal surface. A string is pulled in the direction shown in the figure. In what direction will the yo-yo rotate and move?