title: Dirac representation between matter and anti-matter

text: If $|\psi \rangle$ represents a wavefunction through a column matrix and $\langle \psi |$ represents the dual vector in a row matrix and $|\psi \rangle \langle \psi | = \rho $ is the probability density of the wavefunction collapse to a particular value in $\mathbb{R}$ by a measurement, and matter is also a wave (DeBroglie), how does one represent matter by using Dirac notations - how about anti-matter? Would there be something similar to $\langle \psi |\psi \rangle = n \in \mathbb{R} $ ? I apologize if the question is quite ignorant, i.e. Dirac's equations don't relate matter/anti-matter, it just seems they should - and I would like to know how ;) Thanks, Brian


title: Internal Stresses in a Horizontal Rod

text: Suppose we havean uniform slender rod of dimensions : $L$( length) and $A$( cross sectional area). Now at different points on the rod we apply different forces. So, in this situation how are we expected to find the elongation in the rod? I tried in 2 different ways:   * I considered each part and found the elongation for each part, and then added them up.   * I considered cumulative segments, that is: from part A to B then from part A to C ( hence taking the forces at junction B as internal force and not taking them into account) and then adding them up. My question is, as these two processes yield different results, which is the correct method and why? Also if anyone has a better method, please share it with me. The type of the question can be found in Timoshenko & Young : Elements of Strengths of Materials, Exercise 1.2 figure A.


title: Does the Relativity Principle of Special Relativity imply homogeneity and isotropy of all the reference frames?

text: In Rindler's book: Relativity, Special, General and Cosmological, is stated on page 40 that the Relativity Principle (RP), when applied to just one Inertial Frame (IF), guarantees the homogeneity and isotropy of tha IF. By inertial frame Rindler means an ideal infinity extended rigid body moving freely in a world without gravity. This is distinct from an inertial coordinate system, that should be understood as an IF plus, in it, a choice of standard coordinates $x$, $y$, $z$ and $t$. As he says, the RP concerns inertial coordinate systems: the laws of physics are invariant under a change of inertial coordinate systems. I can't understand why this imply homogeneity and isotropy of an IF. If I suppose the existence of an special direction in some inertial reference frame, I could imagine some physical law governig the propagation of some signal (it can be light if you want, but it's not necessary), and if by measuring the velocity of this signal in two different directions and I get two different results, this would violate the isotropy of the IF and at the same time I could write the physical law in an invariant way under coordinate changes inside de IF (sure, it would depend on the special direction) and this would be in accordance with the RP as stated above. What is wrong with my reasoning?


title: How is Doppler redshift of distant galaxies established?

text: Doppler redshift of distant galaxies gave first hint that the universe is expanding. I am curious to know how this redshift is actually measured and interpreted from observation. Suppose I observe some visible line say red with wavelength \lambda_1. This is not enough to tell that this line is redshifted, i.e., the original line emitted from that galaxy had a smaller wavelength. How do I know the value of that particular line when it was emitted from that galaxy? It appears that mere observation of a line is not enough. So how do we proceed?


title: State space of QFT, CCR and quantization, and the spectrum of a field operator?

text: In the canonical quantization of fields, CCR is postulated as (for scalar boson field ): $$[\phi(x),\pi(y)]=i\delta(x-y)\qquad\qquad(1)$$ in analogy with the ordinary QM commutation relation: $$[x_i,p_j]=i\delta_{ij}\qquad\qquad(2)$$ However, using (2) we could demo the continuum feature of the spectrum of $x_i$, while (1) raises the issue of $\delta(0)$ for the searching of spectrum of $\phi(x)$, so what would be its spectrum?      I guess that the configure space in QFT is the set of all functions of $x$ in $R^3$, so the QFT version of $\langle x'|x\rangle=\delta(x'-x)$ would be $\langle f(x)|g(x)\rangle=\delta[f(x),g(x)]$,   but what does $\delta[f(x),g(x)]$ mean?      If you will say it means $\int Dg(x) F[g(x)]\delta[f(x),g(x)]=F[f(x)]$, then how is the measure $Dg(x)$ defined?      And what is the cardinality of the set $\\{g(x)\\}$? Is the state space of QFT a separable Hilbert sapce also? Then are field operators well defined on this space? Actually if you choose to quantize in an $L^3$ box, many issues will not emerge, but many symmetries cannot be studied in this approximation, such as translation and rotation, so that would not be the standard route,   so I wonder how the rigor is preserved in the formalism in the whole space rather than in a box or cylinder model?      I'm now beginning learning QFT, and know little about the mathematical formulation of QFT, so please help me with these conceptual issues.


title: Hadamard gates and constructing them

text: Can anyone explain the process required to make a Hadamard gate that acts on 1st, 2nd and 3rd qbits? For Hgates acting on the first qubit i realise the matrix is $H=\begin{pmatrix} 1&1\\\1&-1\end{pmatrix}$, but I am unsure how to formulate such a gate. As to acting on the second qubit I have read this web page 1 but I am not really sure if it is correct, because it is just a 4x4 matrix without any zeroes. I found this $\begin{pmatrix} 1 & 1 \\\\-1 & 1 \end{pmatrix} $ on another website which will obviously have zeroes when it is multiplied by $I$. Simply can anyone explain how Hadarmd gates can be formulated and built to n qubits?


title: Electric field in a wire?

text: ![enter image description here](http://i.stack.imgur.com/jHugn.png) The electric field in a wire is parallel to the wire's surface (it is always pointing "to the front"). If the electric field did not point in this direction, surface charges would build up and change the electric field to make it point the correct way. In the above diagram the electric field vectors are pointing the wrong direction, but there will be no charge build up, since the current flowing into the junction is same as the current flowing out. So in this scenario how would the surface charges rearrange to make the electric field point in the correct direction?


title: How is matter stored in black hole?

text: A black hole engulfs the matter nearby, how does it store the mass inside? We know that matter is composed of particles, does a black hole store the mass in massive particles? Or can we assume that it's composed of extremal black holes? what are theories about it? EDIT: I checked out Black-holes are in which state of matter? it doesn't provide a definite answer.


title: Why the quantum entanglement doesn't break quantum cryptography

text: If the Alice sends to Bob an q-bit $\alpha\left|\mathbf{0}\right> \+ \beta\left|\mathbf{1}\right>$ in quantum key exchange (such as BB84) it is assumed that Eve cannot copy the state (due to no cloning theorem) and attempts to read the state requires guessing the base which would allow to detect interception. However if the Eve put $\left|\mathbf{1}\right>$ through C-Not gate than the state would be: $$\phi_0 = (\alpha\left|\mathbf{0}\right> \+ \beta\left|\mathbf{1}\right>)\otimes\left|\mathbf{1}\right>$$ $$\phi_1 = \alpha\left|\mathbf{00}\right> \+ \beta\left|\mathbf{11}\right>$$ Now Eve can store her q-bit somewhere and send the 'original' one to Bob. Since the base disclosure is on public channel she can measure her q-bit in correct base (the state would be collapsed but it shouldn't matter). Why then quantum entanglement doesn't break quantum cryptography? **Edit:** To clarify what I understand (I haven't read the text about density matrices). Alice sends to Bob two qubits - $\left|\mathbf{0}\right>$ and $\left|\mathbf{+}\right>$. Let's assume they are not discarded then in first case after Eve entangle it's qubit with sent qubit the state in case one: $$\phi_{\left|\mathbf{0}\right>} = \left(\begin{array}{rrrr} 0 & 1 & 0 & 0 \\\ 1 & 0 & 0 & 0 \\\ 0 & 0 & 1 & 0 \\\ 0 & 0 & 0 & 1\end{array}\right)\left(\begin{array}{r} 0 \\\ 1 \\\ 0 \\\ 0\end{array}\right) = \left(\begin{array}{r} 1 \\\ 0 \\\ 0 \\\ 0\end{array}\right) = \left|\mathbf{00}\right>$$ Now the only possible value read by Bob is $\mathbf{0}$ assuming he guessed base. Eve can then overhear the disclosure of bases and measure the reduced state on previously stored entangled qubit. In second case: $$\phi_{\left|\mathbf{+}\right>} = \left(\begin{array}{rrrr} 0 & 1 & 0 & 0 \\\ 1 & 0 & 0 & 0 \\\ 0 & 0 & 1 & 0 \\\ 0 & 0 & 0 & 1\end{array}\right)\frac{1}{\sqrt{2}}\left(\begin{array}{r} 0 \\\ 1 \\\ 0 \\\ 1\end{array}\right) = \frac{1}{\sqrt{2}}\left(\begin{array}{r} 1 \\\ 0 \\\ 0 \\\ 1\end{array}\right) = \left|\mathbf{++}\right>$$ Assuming that the Bob guessed the base correctly the Eve would also measured $\mathbf{+}$ in similar way as previously. If Bob didn't guessed it correctly Eve can discard particular qubit. Now if quantum cryptography works:   * For some reasons the entanglement is forbidden (if yes - what is the difference between sending qubit and processing it inside quantum computer?)   * Reduction of state by Bob does not happen in a way I think (possibly due to not knowing the density matrix).


title: Can a simulated Universe simulate Special Relativity

text: I know there are theories (or postulates) that hold that our Universe could be a simulation. I was wondering, if Special Relativity states that two events which are not causally linked can be judged to have happened simultaneously or not, depending on the reference frame of the observer, can someone (or something) simulating the Universe reproduce this effect (if one observer says that two events were simultaneous, and the other disagrees, and they are both equally justified in their claims, what would the simulator conclude regarding their simultaneity)?


title: No-slip condition when one fluid flows through another

text: What sort of no-slip zone is in place when liquid water flows from a faucet through air at room temperature? Does the water drag a layer of air molecules along with it?


title: An intuitive proof of Bertrand's theorem

text: Is there a way to see that Bertrand's theorem is true intuitively. I mean without getting into too much mathematics ?


title: What happens to bodies that lift each other up?

text: `Person A` and `Person B` are same person (theortically). They both weigh 800N. They both can vertically lift 1000N. They want to check something out. `G -> Gravity; N -> "Normal"; M -> Muscular active` `R -> Netto; 1 -> Person A; 2 -> Person B` $F_{g1} = 800N$ $F_{g2} = 800N$ $F_{m1} = 0N$ $F_{m2} = 0N$ $F_{n1} = F_{g1} - F_{m2} = 800N$ $F_{n2} = F_{g2} - F_{m1}= 800N$ $F_{r1} = F_{g1} - F_{n1} - F{m2} = 0N$ (no move) $F_{r2} = F_{g2} - F_{n2} - F{m1} = 0N$ (no move)   * `Person A` grabs `Person B` with both hands shirt of `Person B` in region of a bit lower than collarbone (like "tough" guys do on films).   * `Person A` lifts `Person B` with 1000N in air, about 135 degrees relative to his torso. $F_{g1} = 800N$ $F_{g2} = 800N$ $F_{m1} = 800N$ $F_{m2} = 0N$ $F_{n1} = F_{g1} + F_{g2} = 1600N$ $F_{n2} = F_{g2} - F_{m1} = 0N$ $F_{r1} = F_{g1} - F_{n1} - F{m2} = 0N$ (no move; on ground) $F_{r2} = F_{g2} - F_{n2} - F{m1} = 0N$ (no move; in air)   * Seems till now, perfectly understandable and normal to me. In my perception of physics.   * `Person B` grabs `Person A` with both hands whatever of `Person A` in region of whatever. By whatever I mean, somewhere on body that person is still able to pull `Person A` in air.   * `Person B` lifts(?) `Person A` with 1000N in air(?), about 135 degrees relative to his torso.   * This is when I lose it. Following rules, are total non-sense and also cause why I write this. $F_{g1} = 800N$ $F_{g2} = 800N$ $F_{m1} = 800N$ $F_{m2} = 800N$ $F_{n1} = F_{g1} - F_{m2} = 0N$ $F_{n2} = F_{g2} - F_{m1} = 0N$ $F_{r1} = F_{g1} - F_{n1} - F_{m2} = 0N$ ("floating") $F_{r2} = F_{g2} - F_{n2} - F_{m1} = 0N$ ("floating") "My brain", that gummy moisten tube in my head filled with baloney, tells me that first gravity and muscle force count. Then the normal one. This would cause $F_{r_{both}} = 0N$. They both have effect of being "freed" from effect of "normal" force. As `Person 2` was when he/she was in air, being lifted by `Person 1`. But it's impossible. There is still gravity pulling them to earth. And at this point. There is "floating" going on and explaining this with "Normal force" would make it even worse because both of them separately will be either flying or falling through ground. @edit - When `Person A` will pull `Person B` up. `Person B`'s attitude would decrease till on feet. What then? As if, if they normally stay on ground turning to tomatoes because of pulling each other up. What happens to that all the force they put in it? This 1600N. "Definables": T-Shirt is untearable, won't fall off and does not hurt body at all. Both of them stay right on their feet. Please explain as plain as possible. I am really really really low with my knowledge. Relatively to you, I am idiot. I'm sorry for grammar. **Question:** What happens to `Person 1` and `Person 2`? Could you explain forces influencing their bodies, please? If you are about to think "Goddammit, you're so stupid, answer is so obvious!" keep in mind I am teenager (17) without real physics knowledge. Answer may just lay around my point of view/knowledge/logic. In place of getting offensive try to explain where I made my mistake so it won't happen again :).


title: Why can't Wetterich's “Universe without expansion” be tested compared to standard cosmological redshift?

text: I have read on several sites that this is a theory that cannot be tested. Why is this a theory that can not be tested? It seems to me that if the mass of particles is continually increasing, then the wavelength of a spectral line emitted by an atom would also be continually changing. This means that over a length of time (possibly not human time scale), if we could detect a shift in the emitted wavelength from an atom (stationary compared the observer), then we could determine if this interpretation is correct. We can obviously tell the difference in wavelength between the light emitted 10 billion years ago and that emitted today, so it should just be a matter of how fine our resolution is. The more narrowly we can resolve the wavelength, the shorter time span needed to make the measurement. If red shift is caused by movement relative to the observer, no red shift should be found. If red shift is caused by the mass change over time of the particles, red shift should be observable.


title: What are washers for?

text: When you attach a bolt to something using a nut, it is clear what the roles of the nut and bold are. ![enter image description here](http://i.stack.imgur.com/JC94D.jpg) The more you tighten the bolt the more secure your fastening. However, you are often also told to use a washer as well. ![enter image description here](http://i.stack.imgur.com/JEZK3.jpg) I know this somehow prevents the bolt from loosening but from a physics/mathematics point of view, what is the role of the washer?


title: maximum distance between accelerating objects started at different times

text: Let there be two objects that have zero relative velocity with respect to each other in an inertial frame. If they both undergo identical accelerations, but one starts the acceleration at t = T1 and the other starts the acceleration at t = T2, then per special relativity the maximum distance that rest frame can measure between the two objects is (T1-T2) * c What acceleration equations show that the separation between these two objects as t approaches infinity approaches c * (T1-T2)


title: How should I interpret the expectation value $\langle x p\rangle$ in quantum mechanics?

text: $xp$ is not a hermitian operator and hence doesn't represent an observable. Then, how can we interpret the expression $$ \langle x p \rangle \text{,} $$ the expectation value of position times momentum? How to interpret the expectation value of a hermitian operator, say $p$, is clear: make several measurements on identically prepared systems and get the expectation value from the measured quantities. But now we are talking about the expectation value of an operator that doesn't represent an observable. The discussion above hence isn't valid. However, computing $\langle x p \rangle$ knowing $\Psi$ according to the mathematical definition can be done, so $\langle x p \rangle$ can clearly be deduced if we know the wave function - but how can we interpret the quantity we get? (The question arises from a line in my textbook - "in a stationary state, $\frac{d}{dt} \langle x p \rangle = 0$, ..." without further explanation - why is this obvious?)


title: Liquid crystal polarizes light reflection question

text: I was hoping someone could help me with understanding why a row of polarizes reflects a light wave when the whole row is the same length as the wavelength of the light. ![Liquid crystals](http://i.stack.imgur.com/kdcDG.gif) I pretty much get the physics behind the rest, just don't understand this little part. Why only the light of that wavelength and does this also apply for light coming in at a different phase than the crystals?


title: Which is the smallest known particle that scientists have actually *seen with their eyes*?

text: Which is the smallest particle that has been actually seen by the scientists? When I say "actually seen", (may be using some ultra advanced microscope or any other man made eye, using any wavelength or phenomena) I really mean it; just like we have seen the red blood cells. Davidmh's answer is pretty much in line with what I am asking


title: Standing sound wave tube

text: If there was a standing sound wave tube and a flammable gas was introduced then ignited, would the combustion be more forceful and more efficient since its following a standing wave, than just a gas ignited within a tube?


title: What is weak coupling of photon polarization to a pointer?

text: This question is refered to those who are familiar with the concept of weak measurement. **In short:** How can the polarization of a photon be coupled to the position of a pointer state? What is the pointer state? The position of some particle? How to realise that in a lab? **In detail:** Let's say we want to weakly measure the polarization of a photon. We write the observable $A=|H\rangle\langle H| - |V\rangle\langle V|$ and the initial state shall be $|\psi\rangle = \alpha |H\rangle + \beta |V\rangle$. Now we weakly couple it to a pointer by sending it through a birefringent element, i.e. depending on the polarization the photon needs more time to pass the birefringent element. Formally: The Hamiltonian whichs represents this coupling is given by $H=\lambda A \otimes P$ where $\lambda$ is the coupling strenght and $P$ the momentum operator of the pointer. Now $\exp{(-\text i H t)}$ will act as follows: $|\psi\rangle \otimes |g\rangle = (\alpha |H\rangle + \beta |V\rangle) \otimes |g\rangle \;\longrightarrow\; ( \alpha |H\rangle \otimes|g_+\rangle + \beta |V\rangle\otimes|g_-\rangle)$ where the first degree of freedome is clearly the polarization of the photon and the second is the pointer state (a gaussian say). Both $g_+$ and $g_-$ refere to small shifts due to the birefringent element. Then there is this well known fact that a post-selection whichs leads to an imaginary weak value $A_w$ causes a shift in the momentum-space of the pointer beeing proportional to $\text{Im} A_w$. The weak value is defined as $A_w = \frac{\langle \phi|A|\psi\rangle }{\langle \phi|\psi\rangle}$ where $|\phi\rangle$ is the post-selected polarization state of the photon. (reference: http://arxiv.org/abs/0706.4207 or http://arxiv.org/abs/0911.5139) Mathematically this is easy to understand, but I hace a very conceptual problem with all that: How do I have to understand the pointer state? Is it a position of a particle? If yes, how can its position be coupled (in a physically understandable way) to a photon? And how can a post-selection affect its momentum?


title: Difference between Veneziano amplitude and Virasoro shapiro amplitude

text: I have been study about Veneziano amplitude and Virasoro Shapiro amplitude. I want to summarize this two amplitude in the following way, please check that i am understand them properly. Veneziano amplitude : they are open string amplitude for disk. Usually we called four-open string tachyon on disk as Veneziano amplitude Virasoro-Shapiro amplitude : they are closed string amplitude for sphere. Usually we called four-closed string tachyons on shpere as Virasoro-Shapiro amplitude. The similarity between Veneziano and Viraosoro-Shaprio amplitude are they both have Regge and hard scattering limit. * * * Is there are more difference and similarity for Veneziano and Virasoro-Shaprio amplitude?


title: Does resistance in second circuit changes in transformed circuit?

text: Let $V_1$, $I_1$, $R_1$ and $V_2$, $I_2$, $R_2$ be voltages, currents, resistance in first, second circuit. And we assume $R_1=R_2$, $I_1\neq I_2$. The conservation of electrical power $P=V_1I_1=V_2I_2$ holds in the transformer circuits. Hence $I_1^2R_1=I_2^2R_2$, $R_2=(I_1^2/I_2^2)R_1$ which leads to a contradiction. What am I missing?


Is entanglement necessary for quantum computation? If there was no error in the computation,superposition of states would be sufficient for quantum computation to be carried out.Is this right?


title: Fall/Winter Viewing

text: I live in Seattle and am thinking of purchasing a telescope. Is fall/winter a decent time of year for viewing (aside from summer)? Are there any major viewings/events during that this of year? I know I live in a city of rain, but there are some nice nights.


title: Can the "Milky Way" galaxy be seen by the naked eye in a clear sky?

text: Is this photo "real"? Are the stars not super-imposed in the image? ![galaxy](http://i.stack.imgur.com/BvJZ0.jpg)


title: Are elementary particles actually more elementary than quasiparticles?

text: Quarks and leptons are considered elementary particles, while phonons, holes, and solitons are quasiparticles. In light of emergent phenomena, such as fractionally charged particles in fractional quantum Hall effect and spinon and chargon in spin-charge separation, are elementary particles actually more _elementary_ than quasiparticles? Does the answer simply depend on whether one is adopting a reductionist or emergent point of view?


title: Ignorance in statistical mechanics

text: Consider this penny on my desc. It is a particular piece of metal, well described by statistical mechanics, which assigns to it a state, namely the density matrix $\rho_0=\frac{1}{Z}e^{-\beta H}$ (in the simplest model). This is an operator in a space of functions depending on the coordinates of a huge number $N$ of particles. The **ignorance interpretation of statistical mechanics** , the orthodoxy to which all introductions to statistical mechanics pay lipservice, claims that the density matrix is a description of ignorance, and that the true description should be one in terms of a wave function; any pure state consistent with the density matrix should produce the same macroscopic result. Howewer, it would be very surprising if Nature would change its behavior depending on how much we ignore. Thus the talk about ignorance must have an objective formalizable basis independent of anyones particular ignorant behavior. On the other hand, statistical mechanics _always_ works _exclusively_ with the density matrix (except in the very beginning where it is motivated). Nowhere (except there) one makes any use of the assumption that the density matrix expresses ignorance. Thus it seems to me that the whole concept of ignorance is spurious, a relic of the early days of statistical mechanics. Thus I'd like to invite the defenders of orthodoxy to answer the following questions: (i) Can the claim be checked experimentally that the density matrix (a canonical ensemble, say, which correctly describes a macroscopic system in equilibrium) describes ignorance? \- If yes, how, and whose ignorance? \- If not, why is this ignorance interpretation assumed though nothing at all depends on it? (ii) In a though experiment, suppose Alice and Bob have different amounts of ignorance about a system. Thus Alice's knowledge amounts to a density matrix $\rho_A$, whereas Bob's knowledge amounts to a density matrix $\rho_B$. Given $\rho_A$ and $\rho_B$, how can one check in principle whether Bob's description is consistent with that of Alice? (iii) How does one decide whether a pure state $\psi$ is adequately represented by a statistical mechanics state $\rho_0$? In terms of (ii), assume that Alice knows the true state of the system (according to the ignorance interpretation of statistical mechanics a pure state $\psi$, corresponding to $\rho_A=\psi\psi^*$), whereas Bob only knows the statistical mechanics description, $\rho_B=\rho_0$. Presumably, there should be a kind of quantitative measure $M(\rho_A,\rho_B)\ge 0$ that vanishes when $\rho_A=\rho_B)$ and tells how compatible the two descriptions are. Otherwise, what can it mean that two descriptions are consistent? However, the mathematically natural candidate, the relative entropy (= Kullback-Leibler divergence) $M(\rho_A,\rho_B)$, the trace of $\rho_A\log\frac{\rho_A}{\rho_B}$, [edit: I corrected a sign mistake pointed out in the discussion below] does not work. Indeed, in the situation (iii), $M(\rho_A,\rho_B)$ equals the expectation of $\beta H+\log Z$ in the pure state; this is minimal in the ground state of the Hamiltonian. But this would say that the ground state would be most consistent with the density matrix of any temperature, an unacceptable condition. Edit: After reading the paper http://bayes.wustl.edu/etj/articles/gibbs.paradox.pdf by E.T. Jaynes pointed to in the discussion below, I can make more precise the query in (iii): In the terminology of p.5 there, the density matrix $\rho_0$ represents a macrostate, while each wave function $\psi$ represents a microstate. The question is then: When may (or may not) a microstate $\psi$ be regarded as a macrostate $\rho_0$ without affecting the predictability of the macroscopic observations? In the above case, how do I compute the temperature of the macrostate corresponding to a particular microstate $\psi$ so that the macroscopic behavior is the same - if it is, and which criterion allows me to decide whether (given $\psi$) this approximation is reasonable? An example where it is not reasonable to regard $\psi$ as a canonical ensemble is if $\psi$ represents a composite system made of two pieces of the penny at different temperature. Clearly no canonical ensemble can describe this situation macroscopically correct. Thus the criterion sought must be able to decide between a state representing such a composite system and the state of a penny of uniform temperature, and in the latter case, must give a recipe how to assign a temperature to $\psi$, namely the temperature that nature allows me to measure. The temperature of my penny is determined by Nature, hence must be determined by a microstate that claims to be a complete description of the penny. I have never seen a discussion of such an identification criterion, although they are essential if one want to give the idea - underlying the ignorance interpretation - that a completely specified quantum state must be a pure state. Part of the discussion on this is now at: http://chat.stackexchange.com/rooms/2712/discussion-between-arnold-neumaier- and-nathaniel Edit (March 11, 2012): I accepted Nathaniel's answer as satisfying under the given circumstances, though he forgot to mention a fouth possibility that I prefer; namely that the complete knowledge about a quantum system is in fact described by a density matrix, so that microstates are arbitrary density matrces and a macrostate is simply a density matrix of a special form by which an arbitrary microstate (density matrix) can be well approximated when only macroscopic consequences are of interest. These special density matrices have the form $\rho=e^{-S/k_B}$ with a simple operator $S$ - in the equilibrium case a linear combination of 1, $H$ (and vaiious number operators $N_j$ if conserved), defining the canonical or grand canonical ensemble. This is consistent with all of statistical mechanics, and has the advantage of simplicity and completeness, compared to the ignorance interpretation, which needs the additional qualitative concept of ignorance and with it all sorts of questions that are too imprecise or too difficult to answer.


title: Why are rockets so big?

text: I'm curious why rockets are so big in their size. Since both the gravitational potential one need to overcome in order to put thing into orbit, and the chemical energy burned from the fuel, are proportional to the mass, so if we shrink the rocket size, it would seem to be fine to launch satellites. So why not build small rocket say the size of human? I can imagine small rocket would be easier to manufacture in large quantities and easier to transport. And maybe someone can make a business out of small rocket, carrying one's own satellite.


title: Moduli spaces in string theory vs. soliton theory

text: In both string theory and soliton theory, moduli spaces are frequently used. As far as I known, for soliton theory, moduli spaces are something like collective coordinates for solitons, and for string theory, moduli spaces is the spaces of all metrices divided by all conformal rescalings and diffeomorphisms. It seems like these two definitions(?) of moduli spaces are quite different, but the same terminology is used in both cases. I also learned that the name 'moduli spaces' comes from abstract geometry, but I don't know if that's any help here. My question is the following: Could anyone provide an intuitive connection between the two uses of moduli spaces, or highlight the differences?


title: Reducing Eddy Currents?

text: A conductor that passes a magnetic field, decreases its velocity due to the opposing magnetic forces created by the Eddy Currents. This "magnetic breaking" effect can be reduced if we change the shape of the conductor from a bar, to a this conductor like this: ![enter image description here](http://i.stack.imgur.com/0NFr5.jpg) What I understand is that conductor above has a higher resistance and reduces the eddy currents. However, what if the air gaps are conductors? Here is where I find laminated conductors to be useful like so: ![enter image description here](http://i.stack.imgur.com/cAvF7.gif) However, what is the difference? I believe the laminated conductor can carry more current(i/e higher current capacity) than the one above. The whole point is reducing eddy currents(the magnetic breaking)in electric motors, or generators... and not affecting its electric properties by increasing its resistance and reducing its current capacity. Am I correct here or is my understanding incorrect?


title: Why can't we make measurements in a photon's rest frame when loop diagrams make measurements possible?

text: It is one of the axioms of special relativity that the photon has no rest frame; light travels at speed c when measured in any inertial frame of reference. As a corollary, it is often said that if one _were_ in a photon's rest frame, infinite time-dilation and length-contraction would make the universe appear (from a photon's perspective) to be unchanging, with zero length in its direction of motion. In other words a measurement would not be possible in the reference frame of a photon, because there would be no time or space in which one could conduct it. My question is: how does this accord with the fact that parallel-moving photons can interact with each other via loop diagrams[*]? Photons can therefore be used to perform measurements in the photon's rest frame. I suppose an answer may be "the act of measurement changes the photons momentum vector and therefore its co-moving frame is non-inertial" however this can be said of _any_ measurement. [*] Or can they? I can't find an explicit amplitude to check this (for, say, the photon-photon "box" diagram), however as far as I can tell from the Euler- Heisenberg Lagrangian nonlinear effects should be present for parallel photons.


title: What is a phrase for testing for a certain result?

text: Is there a word or phrase for when someone is testing for a certain result thereby skewing his findings?


title: Largest theoretically possible specific heat capacity?

text: What substance will have the largest specific heat capacity integrated from T=0 to, say, room temperature? In other words, given a finite amount of mass, what object or collection of objects has the largest number of degrees of freedom that can be excited as it absorbs energy starting from T=0? Would it be a complicated molecular polymer that can be tangled in all sorts of ways, or some kind of gas of low-mass particles, or maybe a spin lattice of some sort? Is there some kind of fundamental limit in the universe of the number of quantum degrees of freedom per mass or perhaps per volume that is allowed?


title: What does it mean for a Hamiltonian or system to be gapped or gapless?

text: I've read some papers recently that talk about gapped Hamiltonians or gapless systems, but what does it mean? **Edit:** Is an XX spin chain in a magnetic field gapped? Why or why not?


Does Wick rotation work for quantum gravity? The Euclidean Einstein-Hilbert action isn't bounded from below.


title: Is there a relativistic (quantum) thermodynamics?

text: Does a relativistic version of quantum thermodynamics exist? I.e. in a non- inertial frame of reference, can I, an external observer, calculate quantities like magnetisation within the non-inertial frame? I'd be interested to know if there's a difference between how to treat thermodynamics in a uniformly accelerated reference frame and in a non- uniformly accelerated reference frame. Thanks!


title: What is the need for the Higgs mechanism and electroweak unification?

text: The Higgs mechanism allows massless fields to acquire mass through their coupling to a scalar field. But if the masses cannot be predicted because the couplings have to be fixed, what really is the utility of the Higgs mechanism? Instead of saying "Here are _a priori_ couplings; the Higgs mechanism generates mass.", I could just as well say "Here are _a priori_ masses. Period.". I understand that the Higgs mechanism is crucial to electroweak unification, but I have the same question there. Why does electromagnetism and the weak force have to be unified? Even if the couplings of the photon, Z and W bosons became related on unification, this is still at the cost of introducing new parameters - so it's not really clear to me that something has been explained or tidied up. Do either the Higgs mechanism or electroweak unification tell us something new? Do either make any predictions that don't come at the cost of extra parameters? (I'm not really challenging anything here; I'm sure the answer to both questions is 'yes' - I just want to fill the gaps in my understanding as I study the Standard Model)


title: accelerated charge in QED

text: Classical electromagnetism says an accelerated charge would radiate. Is this understood in QED? Is there QED derivation of Larmor's formula?


title: Beginner projects in NMR quantum computing

text: I have applied for a summer project in NMR quantum computing as I want to learn this field, but my professor wants me to tell him the project title. I have no idea about the field and what projects are good. Please can you suggest me some good 2 month projects. I have done advanced course in quantum mechanics and also I am familiar with NMR machine.


title: Two hanging masses connected by springs

text: I had this problem for a candidacy exam, but wasn't able to get the complete answer. Their spring constants and masses are not the same, find the equilibrium position and frequencies of the system. Apparently once you find their equilibrium position, i.e. when gravity and the spring forces cancel out, you can transform to about that position and ignore gravity. Why do you get to ignore gravity? This is a normal modes problem, but I couldn't justify why gravity just disappears, but it does. Anyone have a simple reason why?


title: Why is the binding energy per nucleon of helium-3 less than that of helium-4?

text: I'd guess it has to do with the structure of helium-3 allowing for greater Coulomb repulsion between the protons, but I'm unsure.


title: What cools a drink?

text: When you stick ice in a drink, AFAICT (the last physics I took was in high school) two things cool the drink:   * The ice, being cooler than the drink, gets heat transferred to it from the drink (Newton's law of cooling). (This continues after it melts, too.)   * The ice melts, and the cool liquid mixes with the drink, which makes the mixture feel cooler. My first question is, to what extent does each of these cool the drink: which has a greater effect, and how much greater?      Secondly, how much of the cooling by the first method (heat transfer) is without melting the ice? That is, is there any significant amount of heat transfer to each speck of ice before it's melted, and how much does that cumulatively affect the drink's temperature?      I suppose all this will depend on a bunch of variables, like the shape and number of ice cubes and various temperatures and volumes. But any light that can be shed, let's say for "typical" situations, would be appreciated.


title: Why is classical mechanics determinism based on position and momentum only and not forces and scattering rules?

text: Consider a closed system (say a box) of $n$ particles. There is a well-known idiom/meme/law in classical mechanics that says that the position and momentum of those $n$ particles is all that is needed to determine the future and past configuration of the system. (A minor question, does this idiom have a name?) Why do we not consider two other pieces of information to be as important as the position and momentum of the $n$ particles?   * The force field (force as a function of space, and preferably of time too).   * The particle interaction rules (When two particles touch each other, do they whiz by or do they scatter? if they scatter what are the rules for determining angles?)


title: How much light can pass through a point?

text: Analogy: an infinite number of lines can pass through a point. Is there a limit on the number of lasers that can pass through a point? Obviously, with lasers the “point” would be a sphere with the diameter of the beams.


title: How do you measure distance to stars within the galaxy?

text: I know that for close by stars (<50 LY) we can use the parallax effect. And for distant galaxies we use red-shift (& hubble's constant). So how do we measure how far is a star lets say 50,000 LY from the earth? I know I am missing something, I just don't know what. How can we assume there is a relationship between red-shift and distance when the stars   * act like a (turbulent) fluid within the galaxy and    * can be moving in any sort of different manner? **Edit:** It is a physics question, because I really want to know what is the star velocity model within the galaxy in order to use red-shift. My instinct tell me that stars move like a swarm within the galaxy with an overall rotation about the galaxy center of gravity (In agreement to the edited horowitz answer).


title: Stress and strain

text: Let us consider a rod having a young's modulus $Y$. Let it be of length $l$, and suppose it is suspended from a point P. Let us pull the rod with a force $F$ at a point Q which is at a distance $2/3l$ from the point P. Next let us apply the same force at the other end point of the rod. Is the length of the rod in both the cases same?


title: Why is the maximum work achieved in reversible processes?

text: Let us consider an ideal gas. Let it be present initially in a state $(p_1,v_1,t_1)$. Now let it be driven to another state $(p_2,v_2,t_2)$. Why is it so that during this process the maximum work can be obtained if the ideal gas is transformed from one state to another by a reversible process ?


title: Quantum Expectation Values

text: I'm having trouble understanding the motivation for the definition of the expectation of a self adjoint operator $A$: $$\langle A \rangle _\psi=\int_{\mathbb{R}}\psi^*A\hspace{0.2cm} \psi \hspace{0.2cm} dx$$ where $\psi(x,t)$ is a normalised state. I can understand the expectation of the position operator in terms of basic probability: one of the assumptions of QM being that $|\psi|^2$ is the probability to find the particle at $x$. The expected value is just the sum of the positions multiplied by their probabilities: $$\langle x \rangle _\psi=\int_{\mathbb{R}}x \hspace{0.1cm} |\psi(x,t)|^2 dx=\int_{\mathbb{R}}\psi^* \hspace{0.2cm} x \hspace{0.1cm} \psi \hspace{0.1cm}dx$$ by commutativity. I don't understand how this might work with the momentum operator, for example. The lecture notes for my course say that the expectation of self adjoint operators are defined in analogy to this, so that $$\langle p \rangle _\psi=\int_{\mathbb{R}}\psi^*\hspace{0.1cm}p \hspace{0.1cm}\psi\hspace{0.1cm} dx$$ I can't satisfy myself with this explanation; $p$ is a differential operator and so can't be moved about within the integral like $x$ can. For example $$\int_{\mathbb{R}}\psi^*\hspace{0.1cm}\psi\hspace{0.1cm} p\hspace{0.1cm} dx$$ makes no sense to me, as the operator hasn't been applied to anything. In any case, $\int_{\mathbb{R}}|\psi(x,t)|^2 p dx$ doesn't mean anything to me probability-wise. So basically I'm looking for an explanation as to why the expectation of self- adjoint operators are so defined. Thanks for any replies!


title: Restrained expansion of a piston/cylinder

text: Suppose an ideal gas in a piston cylinder has some initial pressure $p_1$, volume $V_1$, and temperature $T_1$. A pin, which holds the piston in place, is suddenly removed, and the gas quickly expands. Once equilibrium is reached, the final pressure of the gas is $p_2$. I am looking at a textbook solution which claims that the work done during this process is $$W = \int p\,dV = p_2\left(V_2-V_1\right),$$ which implies that the pressure is constant ($p=p_2$) throughout the whole process, as soon as the pin is released. Is this really the case, and are there not irreversibilities in this process that must be considered?


title: Does back emf occur in transformers?

text: Does back emf occur in transformers and if so does it reduce the efficiency of the transformer and can it be reduced?


title: What equation should be used when this conductor passes a magnetic field?

text: A large thick copper wire(or bar if you like,) passes a magnetic field. What formula is best to calculate the $EMF$ here? $$\epsilon = \frac{\Delta \Phi \ }{\Delta t} $$ or, $$\epsilon = -vBL $$ Example: ![enter image description here](http://i.stack.imgur.com/n8r6i.png) This image is used to demonstrate Lorentz force, however, it can be used to illustrate my question, lets imagine the conductor in the middle is moving and its passing the magnetic field, when it reaches that point in the diagram it is connected to a circuit to allow current to flow.


title: Thermodynamics First Law Question - Expansions

text: > Two thermally iso- lated cylinders, A and B, of equal volume, both equipped > with pistons, are connected by a valve. Initially A has its piston fully > withdrawn and contains a monatomic gas at temperature $T_0$, while B has its > piston fully inserted and the valve is closed. Calculate the final gas > temperature in each cylinder after the following operations, each of which > starts with the same initial configuration. Assume the thermal capacity of > the cylinders is negligible. > > (a) Piston B is fully withdrawn and the valve opened slightly. The gas is > then driven as far as it will go by pushing in piston A at such a rate that > the pressure in A remains constant; the cylinders are in thermal contact. > > (b) As in (a), but the cylinders are thermally isolated from each other. > > (c) Piston B is fully withdrawn and the valve is then opened; the cylinders > are in thermal contact. > > (d) As in (c), but the cylinders are thermally isolated from each other. I think I can do (a) by equating the work done (i.e. due to the piston moving in) to the change in internal energy - but it is (b) when they are thermal isolated that I am stuck on how to approach (b) .


title: Classical rod-wall-floor system

text: I have a homogeneous thin rod $AB$. $A$ can slide along $z$-axis, $B$ along $xy$ plane. There is no friction. We have any initial conditions. Now the question is to write down the equation of motion of the rod. It is assumed that I don't know Lagrangian mechanics. I believe I should write some conservation theorem but I'm not quite sure how to do it. I guess I could finish that problem if I would get some direction.


title: Truck driven from a small motor that can carry a heavy load, yet can travel fast.

text: I am building a truck from trash as materials. I have one small motor and a a few small gears, but no other engineered materials are allowed. The truck must carry a load for a distance of 3m. The winning truck is the truck with the greatest ratio of load carried:time. What do you think would be the best approach to this project in terms of the gears and the radius of the driven wheels? I don't know what would be the best approach in terms of torque and angular velocity. Any other tips or inputs would be greatly appreciated! Thanks!


title: Electroweak Symmetry Breaking

text: Electroweak symmetry breaking is manifested by the nonzero masses of the weak W and Z bosons, and requires the existence of a Higgs boson or some other yet unseen mechanism. To solve the problem of flavor.   * why there are six quarks and six leptons forming three families?   * why these particles exhibit such a peculiar pattern of masses and mixing?


title: Why must the gravitational wave components be much less than unity?

text: We start with the metric tensor \begin{equation} g_{\mu\nu}(x) = \eta_{\mu\nu} + h_{\mu\nu}(x) \end{equation} in the linearised theory, or \begin{equation} g_{\mu\nu}(x) = \bar{g}_{\mu\nu}(x) + h_{\mu\nu}(x) \end{equation} in the more general case with a curved background. However, in both cases, we apparently need \begin{equation} \vert h_{\mu\nu}\vert \ll 1. \end{equation} This condition demands that the components of the perturbation are much less than 1, in the coordinate system where the components of the background metric tensor are of order 1. Why do we need this condition?


title: If helicity can suppress weak decay of $\pi^{\pm}$, doesn't that make chiralty observer dependent?

text: I have been under the impression that chirality is at least in part a way to sneak in a Lorentz invariant version of helicity for particles that mass. Flip Tanedo seems to espouse this view in a somewhat cutesy fashion, for example, though he seems to be using the math rightly. ( _Mea culpa_ BTW, with cute little bee-like particles $e$lly $p$eter $p$aul and $\mu$ary in fact.) Tanedo makes this observation: > ... is there some property which is equivalent to helicity in the massless > limit [and] all observers in valid reference frames would measure to be the > same for a given particle[?] The good news is that such a property exists; > it is called chirality. So, having convinced myself that chirality must somehow manage to remain experimentally independent of helicity -- since helicity _is_ of course frame dependent -- imagine my surprise when I noticed Michael Brown's comment about helicity suppressing weak decays of the $\pi^{\pm}$: > Indeed, you see "helicity" suppressed weak decays of the $\pi^{\pm}$. You > could just as well call it chirality suppression since what is being > measured is the rotation between chirality and helicity eigenstates which is > the result of the mass term. Eh? Someone help me with this: Even if the various constituent components of such decays are _by themselves_ Lorentz independent, what good does that do if their relative ratios are dependent on helicity, which would then as best I can understand it would again make the decay rate of $\pi^{\pm}$ decay frame dependent? I'm sure I'm missing something here, but I honestly don't know what. I've been looking for the papers on the original experiments, but I was hoping someone could point out where my understanding took an, um, left-hand turn...


title: Is there a hard upper bound to the deBroglie wavelength of a particle with vanishing momentum?

text: This is probably a stupid and simple question, but does the heisenberg uncertainty principle set this upper bound? That knowledge of the momentum is limited, so it can't reach a very low value and thus have a very large (visible) debroglie wavelength? I am figuring if you wanted to test this on a macroscopic particle you would need it frozen to near-absolute zero, but I assume QM comes in and makes it impossible to see visible wavelengths with the naked eye. Thanks!


title: Can a macroscopic body have wavelength as that of electron?

text: * Einstein has suggested that light can behave as a wave as well as like a particle i.e, it has dual character. In 1924, de-Broglie suggested that just as light exhibits wave and particle properties, all microscopic material particles such as electrons, protons , atoms, molecules, etc. also have dual character. They behave as a particle as well as a wave. This means that an electron which has been regarded as a particle also behaves like a wave. Thus, according to de-Broglie,  > _all the material particles in motion possess wave characteristics._ According to de-Broglie, the wavelength associated with a particle of mass m, moving with velocity v is given by the relation,   **$$\lambda=\frac{h}{mv}=\frac{h}{p}$$**   where h is Planck's constant, v is the velocity and p(=mv) is momentum of the particles. The waves associated with material pariticles are called de Broglie waves.   * My book says that:  > "Although the dual nature of matter is applicable to all material objects > but it is significant for microscopic bodies only. For large bodies, the > wavelengths of the associated waves are very small and cannot be measured by > any of the available methods. Therefore, practically these bodies are said > to have no wavelengths. Thus, any material body in motion can have > wavelength but it is measurable or significant only for microscopic bodies > such as electron, proton, atom or molecule. This may be illustrated as > follows:   >  The wavelength of an electron with mass $9.11*10^{-31}$kg and moving with > the velocity of $10^6$ m/s is $7.28$ m as shown below:   >  $$\lambda=\frac{h}{mv}=\frac{6.63*10^{-34}kg m^2 > s^{-1}}{(9.11*10^{-31}kg)(10^6m/s)}=7.28*10^{-10}m$$   >  This wavelength associated with the moving electron is of the same order of > magnitude as of $X$-rays which can be easily measured."   * I made an attempt to check the wavelength associated with a car of mass $10^{6}$ kg and moving with velocity of $9.11*10^{-31}$ m/s, and I got the wavelength associated with it as shown below:   $$\lambda=\frac{h}{mv}=\frac{6.63*10^{-34}kg m^2 s^{-1}}{(10^6kg)(9.11*10^{-31}m/s)}=7.28*10^{-10}m$$   This shows that a car or any material object with mass $10^{6}$kg and moving with velocity $9.11*10^{-31}$m/s (almost at rest) has the wavelength same as that of an electron, but this result in contradiction with the statement of my book. And even I thought that it is practically impossible to see any wavelength associated with such mass of $10^{6}$kg. I have heard about rest and relativistic mass, and theory of relativity, but I have naive idea about it. If I have made mistake there, correct me. Or else " _is it that macroscopic body can have same wavelength as that associated with an electron?_ " Please explain.


title: What will be the shape states of water in a centrifuge in 0G

text: Consider the concept of these videos http://www.youtube.com/watch?v=Zip9ft1PgV0&sns=em and http://www.youtube.com/watch?v=cUhgKFV5Ri4&sns=em but set in a zero gravity environment and in a fully encapsulated and high pressure container. My questions are as follows: What shapes will the water conform to at increasing speeds? Will the water conform to a parabola and then ultimately a hollow cylinder as it does on earth? Is it possible that without an opening to spit and by continuously adding more water to the centrifuge, that the water could ultimately fold in on itself and conform to various states of torus, by creating vortices within itself? [Edit] Also, taking into account 0G, will the water even conform to its bounding container in a manner in which it receives traction and can be spun?


title: Calculating the amount of heat energy radiated by sun

text: How do I calaculate the amount of heat energy radiated from the sun in one minute?? Well, i tried some stefan method, but the answer seems far off from the correct one....well, i would thus strongly want to have some help in this regard..


title: Problems in the modern semiconductor/electronics technology?

text: From what I have read, the problem with modern semiconductors/electronics seems to be quantum tunnelling and heat. The root of these problems is the size of the devices. The electrons are leaking out, and currents are causing active materials to melt. How far have we become in this regard? Can we make our devices even smaller? What is being done to maintain advancements in computing power? What is the main research, particularly in quantum mechanics and in solid state physics, being done to compute faster using less energy and space?


title: Induced "Potential Difference" in Loop/How do Voltmeters work?

text: Say we have a single, solid circular loop of wire rotating at a constant angular velocity in a uniform magnetic field. There is an induced EMF creating a current in the wire. _At the absolute minimum_ , we should be able to say that the potential V is 2pi periodic (it should have a single value at a single position on the loop). In my understanding of the problem, the wire ought to be equipotential, even. While the wire is radiating energy due to its resistance, that energy is getting put back into the system by whatever is spinning it at constant angular velocity. Now, break the loop and attach a voltmeter to the two ends of the wire. It will read a potential drop. I am struggling to rectify these two ideas. Help? (I know there are some similar questions to this, but I have not seen an answer that specifically addresses this issue.)


title: electron pathways

text: if there are two identically equal resistance paths, and you have a large group of electrons, will half of the electrons travel lets say left and half right? because I've been thinking that maybe the electrons travel down lets say the left one till there is a large enough amount of electrons that causes the resistance to change by just a fraction, thus forcing the remainder to switch and go right, and so on till all the electrons have traveled.. is this correct?


title: How does electricity 'decide' on it's pathway?

text: I'm struggling to understand the fundamental concepts of electricity, more specifically, the way in which it 'chooses' its optimal pathway. I appreciate electricity will always choose the path of least resistance, but how does it know which route has least resistance? Does it attempt that, potentially sub-optimal, route first before deciding not to take it? This question comes from a discussion on the effect of electricity on the body when hanging on a charged wire, as opposed to hanging on a charged wire whilst touching the ground. I can hang on the wire all day, safely, but the minute I touch the ground the electricity all of a sudden decides that that is the best route. Did it previously try this route?


title: Electricity takes the path of least resistance?

text: > _Electricity takes the path of least resistance!_ Is this statement correct? If so, why is it the case? If there are two paths available, and one, for example, has a resistor, why would the current run through the other path only, and not both?


title: What is happening over the 15 minutes it takes a neutron to decay?

text: I've read that free neutrons decay into a proton, electron and neutrino with an average lifespan of about 15 minutes. Is there anything physically different about a neutron that has existed for 14 minutes and one that has only existed for one minute? Does a random outside event trigger the decay or is something internally in the neutron slowly falling apart?


title: Height of the atmosphere - conflicting answers

text: Okay. I have two ways of working out the height of the atmosphere from pressure, and they give different answers. Could someone please explain which one is wrong and why? (assuming the density is constant throughout the atmosphere) 1) $P=h \rho g$, $\frac{P}{\rho g} = h = \frac{1.01\times 10^5}{1.2\times9.81} = 8600m$ 2) Pressure acts over SA of Earth. Let r be the radius of the Earth. Area of the Earth is $4 \pi r^2$ Volume of the atmosphere is the volume of a sphere with radius $(h+r)$ minus the volume of a sphere with radius $r$. $\frac{4}{3}\pi (h+r)^3 - \frac{4}{3}\pi r^3$ Pressure exerted by the mass of the atmosphere is: $P=\frac{F}{A}$ $PA=mg$ $4\pi r^2 P = \rho V g$ $4\pi r^2 P = \rho g (\frac{4}{3}\pi (h+r)^3 - \frac{4}{3}\pi r^3)$ $\frac{4\pi r^2 P}{\rho g} = \frac{4}{3}\pi (h+r)^3 - \frac{4}{3}\pi r^3$ $3 \times \frac{r^2 P}{\rho g} = (h+r)^3 - r^3$ $3 \times \frac{r^2 P}{\rho g} + r^3 = (h+r)^3$ $(3 \times \frac{r^2 P}{\rho g} + r^3)^{\frac{1}{3}} - r = h$ $(3 \times \frac{(6400\times10^3)^2 \times 1.01 \times 10^5}{1.23 \times 9.81} + (6400\times10^3)^3)^{\frac{1}{3}} - (6400\times10^3) = h = 8570m$ I know that from Occams razor the first is the right one, but surely since $h\rho g$ comes from considering the weight on the fluid above say a 1m^2 square, considering the weight of the atmosphere above a sphere should give the same answer?


title: Spatial part of wavefunction of helium, both electrons are in ground state

text: If both electrons are in ground state, characterized by $(1s)^2$, this means that n =1, l = 0. The two electrons occupy the same energy and space (why doesn't pauli exclusion principle prevent this?), so the spatial part is symmetric. So in order for the state to obey fermi-dirac statistics, the overall wavefunction (spatial*spin) must be anti-symmetric. Does this mean that the spin part is necessarily anti-symmetric? Thus it must be a tripplet state.


title: Empty batteries magically resurrect after reinserting them

text: It happened to me quite often (most recently with my wireless keyboard) that a battery stopped to work and then, if I unplug it and then plug it back in it works again, not just for a couple of minutes but even for a day or two, then you can repeat the process but this time it will last less and so on until it finally dies. The same trick works for my tv control. I can't really understand why, can somebody help?


title: Why are breeder reactors not more prevalent?

text: I recently learned about breeder reactors, which are able to create more fissile material than they consume. They seem to be superior to the types of nuclear reactors currently in use. What are the reasons why breeder reactors are not more prevalent.


title: Charge distribution on a plate of Capacitor with Dielectrics .

text: I had this Homework Problem with a capacitor (parallel plate) that has a group of 3 dielectrics between it like so : ![Descriptive Image](http://i.stack.imgur.com/WbQx2.jpg) Now We were asked to find the equivalent capacitance and the distance of separation between the plates was $d$ and its area was $A$. Now i assumed that the 3 individual dielectrics would act as individual capacitors and K2 would be parallel to K3 and their resultant series with K1 This was also the method I found in many other books. $$C_{k1}~=~ \frac{2K_1\epsilon_oA}{d} $$ $$C_{k2}~=~ \frac{K_2\epsilon_oA}{d}$$ $$C_{k3}~=~ \frac{K_3\epsilon_oA}{d}$$ Finally $$C_{2,3}= \frac{K_2\epsilon_oA}{d} + \frac{K_3\epsilon_oA}{d} \implies \frac{(K_2+K_3)\epsilon_oA}{d}$$ And $$C_{(2,3),1} = \frac{\frac{(K_2+K_3)\epsilon_oA}{d}.\frac{2K_1\epsilon_oA}{d}}{\frac{(K_2+K_3)\epsilon_oA}{d}+\frac{2K_1\epsilon_oA}{d}}$$ Our teach told us that the answer was incorrect , When I told him about the books he told us that the books had it all wrong. So I asked him for the solution . He divided the First Dielectric Into two parts along the line joining the bisector of area : ![enter image description here](http://i.stack.imgur.com/eWBHL.jpg) And Did the shown Charge distribution. Later he equated the potentials and stuff like so $$V_{Q'_1,Q'_2}=\frac{Q'd}{K_1\epsilon_oA}+\frac{Q'd}{K_2\epsilon_oA}~~~~....1$$ $$V_{Q''_1,Q''_3}=\frac{Q''d}{K_1\epsilon_oA}+\frac{Q''d}{K_3\epsilon_oA}~~~~....2$$ then said that $V_{Q'_1,Q'_2}=V_{Q''_1,Q''_3}$ he then put the values of $Q',Q''$ in the equation $$C=\frac{Q'+Q''}{V}$$ Basically getting a result that was quite different from the result given in books .   * I asked him how a single plate , the one in contact with the K1 dielectric could have Two different charges on the same surface.   * He told me that it was because of the accompanying dielectrics    * Was my question valid the first point ?   * Whose solution is correct?


title: Why isn't Hawking radiation frozen on the boundary, like in-falling matter?

text: From the perspective of a far-away observer, matter falling into a black hole never crosses the boundary. Why doesn't a basic symmetry argument prove that Hawking radiation is therefore also frozen on the boundary, and therefore not observable? Wouldn't the hawking radiation have to have started its journey before the formation of the black hole? Furthermore, wouldn't the radiation be infinitely red-shifted?


title: Is it possible for a black hole to form for an observer at spatial infinity?

text: To my knowledge if you calculate the coordinate time (time experienced by an observer at spatial infinity) it takes an infinite amount of time for an object to fall past the horizon of a Schwarzschild black hole. Doesn't this imply that it takes an infinite amount of coordinate time for a Schwarzschild Black Hole to form since the last bit of in-falling matter won't ever ross the horizon as observed by someone at spatial infinity? If so, is it possible for other types of black holes (Kerr etc.) to form in finite coordinate time?


title: Is it intuitive that the conserved quantity from time symmetry is what we know as energy?

text: Is there an easy (aka intuitive) way to understand that the conserved quantity from time translation symmetry is just what we call energy? In other words, we use two definitions of energy. One is with Noethers theorem, and I've been told this is the fundamental one. The other is what is you learn in school and is mentioned in the examples below. The question is how to connect this two definitions. Examples   * I can lift weights, so they get more energy.   * I can boil water for tee, so it gets more energy.   * I can burn $CaO$ with carbon, so I get carbide, that has more energy.   * .. If I define energy as conserved quantity, how do I arrive at my examples..? (Well, "easy/intuitive" is in the eye of the beholder. Thank you nevertheless.) Side question: We have energy conservation in thermodynamics. I have never seen a Lagrangian formulation of thermodynamics. Can I only hope for an answer of my main question in theories that have such a formulation?


title: Feynman Diagrams: Mass Conservation

text: Can a Feynman diagram end with more matter than it began with? The answer should be no because it would break the law of conservation of mass, right? Example: I learned that two up-quarks could interact to create a Higgs boson (via what I learned as "vector fusion"). If this new Higgs were to decay into an up-quark pair, wouldn't the new up-quark be able to interact with the old one to produce another Higgs, creating a potentially infinite amount of particles? I'm new to this site (and physics), so if this is unclear, please comment.


title: Anharmonic oscillators: why is $F=-k x-k' x^3$, with no quadratic terms?

text: The equation of motion of a general anharmonic oscillator includes a position- dependent force that can be expanded in a Taylor series as $$m\ddot{x}+2\mu\dot{x}+k_0+k_1x+k_2x^2+k_3x^3\ldots=F.$$ I am considering a case with force $F$ higher than the critical amplitude, which means I'm in the Duffing regime. The Duffing equation is $$m\ddot{x}+2\mu\dot{x}+k_1x+k_3x^3=F,$$ where the third order term is the Duffing perturbation to the simple harmonic oscillator. This destroys the symmetry around the resonance of, say, amplitude plotted against frequency. I get that we drop the $k_0$ term with a translation to the equilibrium point. However, I don't get why the second order term disappears. The books I've read (in particular Cleland's _Foundations of Nanomechanics_ and Radons' _Nonlinear Dynamics of Nanosystems_) say it's because the harmonic potential is quadratic, hence symmetrical, and hence the sign of $x$ doesn't matter, but that isn't very clear to me; it's not making click. It's probably something very basic but maybe someone can explain it to me simply and briefly? I've also read that even-ordered nonlinearities only mix signals to DC and twice the drive frequency, as opposed to odd-ordered ones which mix back to the operating frequency band, but I don't quite understand the use of the word "mix" in that context.


title: Liquid nitrogen condensing oxygen out of the air

text: In this video, a pool of liquid nitrogen in a metal bowl can cool it enough so that oxygen from the air condenses at the bottom: http://www.youtube.com/watch?v=Ojl10zMCYkY&list=PLmyvBUHJ4pVC3FrSXsT_5S7dFMKPLG_pS&index=1 Thermodynamically, it seems that the bowl can't get cold enough. The bowl can't cool off below the temperature at which the nitrogen pressure in the air (80kPa, I got all the vapor pressure information form Air Liquide) equals the vapor pressure of the liquid nitrogen; at that point the rate of evaporation equals the rate of condensation from the pool. This happens at ~76K. This is also the threshold where oxygen can condense (vapor pressure of 20kPa). However, 76K is a lower bound. The actual temperature will be higher when you include condensation of water into/onto the bowl, convective/radiative heat transfer, and the fact that there is additional nitrogen/oxygen above the surface of the pool beyond the 80kPa used in the calculations. In the face of these calculations, why does it seem to work? My reasoning is that it still gets cold enough for a _mixture_ of nitrogen and oxygen to condense which has enough oxygen to burn the tissue paper, is this correct?


title: Ice and liquid water interacting across a boundary

text: Imagine we have two thermodynamic systems, one a mass of ice and the other an equal mass of liquid water, with both at 273.16K. Each system is isolated, except that they can interact with each other across a boundary that permits the exchange of heat but not matter or work. What will the two systems look like at equilibrium? Somehow I want to automatically imagine that each system will be identical, a combination of liquid water, ice, and water vapor at 273.16K. But if this is true then the two systems were initially at the same temperature but not in thermodynamic equilibrium, an apparent violation of the zeroth law of thermodynamics.


title: Reference request for low entropy big bang

text: There is a somewhat widely accepted argument that the second law of thermodynamics exists because the universe began in a low-entropy state. I'm writing a paper that mentions this (and must be finished today), but I've realised that I don't know who originated this idea. So does anyone know the original reference for this? **Progress** My paper is now submitted, but I'm still curious about the origins of this idea. @Sundak below gives the hint that Boltzmann might have been its originator. Based on this and a bit of searching, I was able to turn up his Lectures on Gas Theory, §90, in which he says "That in nature the transition from a probable to an improbable state does not take place as often as the converse, can be assumed by assuming a very improbable intial state of the entire universe surrounding us...", which sounds very much like the low- entropy big bang idea. However, what he's actually suggesting is that our local region of the Universe had fluctuated into a low-entropy state at some point in the past. He proposes this because he believes the Universe to be infinite in both time and space, implying that the universe as a whole must be in thermal equilibrium. He then goes on to say that at "distances $10^{10^{10}}$ times the distance of Sirius" there might be other civilisations for whom the arrow of time points in the opposite direction. Boltzmann is a fun guy to read. But the idea he's proposing here is not the same as the modern notion that the Universe simply started out in a low entropy state. Skipping forward many decades, in the abstract for his 1979 paper Singularities and Time's Arrow, Roger Penrose describes the attribution of the second law to the universe's boundary conditions as being part of "the normal point of view". I suspect that the references in Penrose's paper might be a good place to look, but it seems that only the abstract is available online. So that narrows it down to some time between 1896 and 1979, but probably towards the beginning of that period. It could well be that Boltzmann himself proposed the idea in a different work. If anyone has any further insight it'd be much appreciated.


title: Particle In a Box and momentum, velocity

text: So on a homework assignment, we are give the width of a well, $a$, and the mass of the particle $m$ and we want to find the average velocity of the particle at the n=1 state. So here is my attempt at the solution: Using, $\psi (x) = \sqrt{\frac{2}{a}} sin(\frac{\pi x}{a})$, I was able to integrate $$\langle p^2\rangle=\int_0^adx\psi^*(x) (-i)^2\hbar ^2\frac{d^2}{dx^2} \psi (x)$$ which yielded $$\langle p^2 \rangle = \frac{\hbar ^2\pi ^2}{a^2}$$ I'm stuck on the remaining parts. So the problem gives us a hint, to use the relation $p^2 = (mv)^2$. I'm not sure how I would go about finding average velocity from $\langle p^2\rangle$, since $\langle v^2\rangle$ and $\langle v\rangle^2$ are not equivalent. And according to the problem, the speed is supposed to come from the kinetic energy, which I don't understand (I derived average KE by using $\langle p^2 \rangle$, so I don't understand why finding KE is necessary?). This last part has been giving me trouble, and I would really appreciate any help.


title: Why is the functional integral of a functional derivative zero?

text: I'm reading Quantum Field Theory and Critical Phenomena, 4th ed., by Zinn- Justin and on page 154 I came across the statement that the functional integral of a functional derivative is zero, i.e. $$\int [d\phi ]\frac{\delta F[\phi]}{\delta\phi^{\alpha}(x)} = 0$$ for any functional $F[\phi ]$. I would be most thankful if you could provide a mathematical proof for this identity.


title: Dingle vs. Bondi: Twin Paradox Debate on BBC radio?

text: Herbert Dingle and H. Bondi debated the twin paradox on BBC radio before 1971. Does anyone have a link to the audio of this debate? thanks


title: Boundary conditions in Electrostatics

text: If I have a grounded conducting material, then I know that $\phi=0$ inside this material, no matter what the electric configuration in the surrounding will be. Now I have a conducting material that is not grounded, then there will be (as long as I am dealing with static problems) no electric field inside this material. Therefore the potential will be constant inside this material, right? Question 1:Therefore, is there any difference in the boundary conditions if I am dealing with a grounded conducting material and an insulator around or a non-charged insulated conducting material and an insulator around? Question 2:Is it possible to get a non-zero potential inside an uncharged insulated conducting material? Especially, would you get a non-zero potential inside a conducting insulated material due to image charges? Question 3:Of course, I read a few pages in Jackson's book about this and saw that he substituted the problem of a charged insulated conducting sphere in an external field with the one of having a grounded conducting sphere in the external field that has a charge sitting in the center of the sphere. Then, the magnitude of the extra charge was given by the difference of the initial charge of the sphere minus the induced image charge on the grounded conducting sphere. Is it possible to make a general substitution like this: Thereby I mean, that we substitute a charged insulated conducting material carrying a charge by a grounded conducting material that has an additional charge(magnitude given by the difference of total charge-image charge) sitting on its surface? So, I would solve the grounded problem and would add the difference of the total charge-image charge to the surface of the material and add this field to the field calculated for the problem of the grounded material.


title: The definition of "frequency" in different contexts

text: I have been doing some research on all kinds of sound-related topics lately and have been a bit confused by the different uses of the term "frequency". Of course, the most general meaning of frequency is just how many times something happens during a certain period of time (a second, usually). However, I've also come across many texts referring to a sound containing multiple frequencies, which seemed weird to me at first. Now, after reading up on stuff like Fourier analysis I've come to understand that "frequencies" in this case refers to the frequencies of the sine waves that make up the sound. So I guess my question comes down to this:   1. Is the above correct, or am I still not understanding it correctly in some way?   2. Assuming it is, is this not pretty ambiguous terminology? You could for example say that a square wave with a frequency of 440 Hz (meaning the waveform repeats 440 times per second) consists of many different frequencies (meaning it consists of many sine waves with different frequencies). It's not hard to see how this could be confusing.   3. And in that case, why can I not easily find a clear explanation of this anywhere? It took me quite some time to piece this together myself and seems like it really should be pretty basic stuff. For example, I first really started thinking about this when I ran into the Nyquist frequency. It seemed logical to me that you needed at least two samples per period to represent any kind of change, so the maximum frequency of the signal could only be half of the sampling rate, but when I started involving different kind of waveforms I realized there would be no way to distinguish between them. It took me a lot of time to figure out that the Nyquist frequency implicitly refers to the maximum frequency that any sine wave present in the signal could have, making it impossible to sample any other waveform at the Nyquist frequency. Of course, if anyone could explain or point me in the direction of some material explaining the importance of sine waves in a more general fashion, and why these things seem to be supposed to be so obvious, that would be appreciated, but the three questions above are what I'm really wondering about.


title: Are Zeppelins still not feasible?

text: As far as i know Zeppelins were abandoned because they were not safe, back in time. Is it still the case in 2011 though? In my opinion it is more economical to operate a Zeppelin than a plane and they are more stable as there is less acceleration so the interior has tables and chairs etc.


title: Black hole complementarity

text: Black hole complementarity states that two observers, one falling into a black hole, and one observing outside, experience two different histories but since they can not communicate there is no inherent contradiction..imagine an observer far from a black hole watching some matter fall into a black hole. The observer sees the matter freeze at the horizon, but the matter itself does fall in. Now, if you measure the mass of the black hole before and after the matter falls in, would you not measure a higher mass? And wouldn't this violate the principal of complementarity? You don't see the matter fall in, yet you see the black holes mass increase. Help?


title: Why does energy in earthquake waves seem to go up with the three halves power?

text: My question might be based on a false premise, so here's why I asked. If you look up the meaning of the moment magnitude scale for measuring earthquake size, the articles usually say that each increase in magnitude by 1 point represents a tenfold increase in the amplitude of the wave, but a 31.6 increase in energy. Which suggests the energy is going up as the three halves power. With other waves the energy goes up with the square of the amplitude, so what is going on here?


title: What is a wave?

text: I was watching an outtake of Prof. Brian Cox talking to a tv producer about "gravity waves". Their discussion got a bit side-tracked, because the non- scientist didn't seem to understand what a wave "would look like", and also because there seemed to be some difficulties with the phrase "moving through", with Prof Cox saying that a wave doesn't move through space-time and then saying the ripples do. I'm now not sure what moves when a speaker plays a sound; the air vibrates, but does "ripple outward"? So, what is a wave? Obviously, I'm really after a much simplified, probably broken, metaphor that allows me to visualise what a wave is. I understand that it will have flaws. I forgot to include the youtube link, so here it is: http://www.youtube.com/watch?v=6EJ8r9Rhfw8


title: refractive index of air in dependence of temperature

text: What is the exact dependence of the refractive index of air and the temperature? Is there a theoretical derivation of it?


title: Does "sea level temperature" exist?

text: When reporting barometric pressure, weather stations normalize the result to sea level using a well-known formula (approximately, add 1 inch to the measured pressure for every 1000 feet above sea level, but I believe the formula they use is more precise). Does a similar concept exist for temperature? As altitude increases, temperature normally decreases. Is there a way to compensate for this effect? Reason: I often notice "unusually cold" spots on temperature maps, which are usually the result of high-altitude stations. It'd be nice to see what the temperature would look like if all stations were at sea level.


title: Why doesn't dark matter affect planetary motion?

text: If the universe is made up of ~95% dark matter, and it interacts only gravitationally then why didn't Newton and Kepler discover it before ? Why does it show itself only in the radial velocity profile of stars in galaxies and not in that of planets around the Sun ?


title: Finding power with Drag Force equation

text: The mass of the car is 1500 kg. The shape of the body is such that its aerodynamic drag coefficient is $C_D=0.330$ and the frontal area is $2.50 m^2$. Assuming that the drag force is proportional to $v^2$ and neglecting other sources of friction, calculate the power required to maintain a speed of $100 km/h$ as the car climbs a long hill sloping is $3.20^\circ$.   Use the formula: $$ F_D=\frac{1}{2}C_D\rho_{air}Av^2 $$ where $\rho_{air}=1.2kg/m^3$ I don't know how to incorporate power into the formula. I get that $F_d$ is proportional to $v^2$ so am I able to cancel them out?


title: How to solve the heat equation for compound materials with different heat conductivities numerically?

text: I'm solving the heat equation with time dependent boundary conditions numerically in a 2D system using the ADI scheme. For the purpose of this question, let's assume a constant heat conductivity and assume a 1D system, so $$ \rho c_p \frac{\partial T}{\partial t} = \lambda \frac{\partial^2 T}{\partial x^2}. $$ This works very well, but now I'm trying to introduce a second material. This one differs slightly in heat capacity and density but has a very different heat conductivity and is connected to the other material by a sharp interface, i.e. a stepwise change in $\lambda$. How should this be treated numerically in the ADI scheme? I can think of different approaches:   1. Treat the two materials as independent domains and connect them by a boundary condition calculating the heat flow in and out of the interface in terms of temperature on the other side of the interface in the last time step. Use a simple forward difference for that on both sides of the interface.   2. Treat it as one domain and use a very fine discretization close the interface as compared to the homogeneous material. Use a scheme like $$ \lambda_{left} \frac{T_i - T_{i-1}}{\Delta x_i} = \lambda_{right} \frac{T_j - T_{j+1}}{\Delta x_j}, $$ where $i$ and $j$ are the points left and right of the interface, instead of the the standard ADI for those points.   3. Drop the assumption of constant heat conductivity and use $$ \rho c_p \frac{\partial T}{\partial t} = \frac{\partial \lambda}{\partial x}\frac{\partial T}{\partial x} + \lambda \frac{\partial^2 T}{\partial x^2}. $$ But in order do so one needs to approximate derivative of lambda at the step position, i.e. introduce an unknown characteristic width $s$ of the **sharp** interface. I assume, that the (more or less arbitrary) choice of this width will significantly influence the system's behaviour. Any advice?


title: Normalization of $\langle p_1 p_2 \vert p\rangle$ in RelQM and NonRelQM

text: Suppose a particle p of three momentum $\vec p$ decays into two particles of 3-momentum $\vec p_1$ and $\vec p_2$. I know the question might sound stupid but right now my brain is full stop: Is the following "ok" $$\langle \vec p_1, \vec p_2 \vert \vec p\rangle = (2\pi)^3 \delta^3(\vec p-\vec p_1-\vec p_2)?$$ If one wants a Lorentz invariant normalization one should use (see eg Srednicki 2011, eq. 9.3) now four vectors! $$\langle p_1 p_2 \vert p\rangle = 2p_0 (2\pi)^3 \delta^3(\vec p-\vec p_1-\vec p_2)$$ but my question then is how this translates to nonrelativistic physics/normalizations? Or perhaps there can't be a nonrelativistic limit of this because in ordinary quantum mechanics particle number is fixed?


title: Infinite degeneracy

text: Is something special for a quantum system with infinite degeneracy like free particle levels? $E=\frac{\hbar^2 \vec{k}.\vec{k}}{2m}$ Edit: I mean what is physical (or mathematical) significance of infinite degeneracy?


title: Multiparticle generalization of $\langle \vec k \vert E,l,m \rangle$ spherical harmonics.

text: From Sakurai eq. 6.4.21a we have that $$\langle {\bf k} \vert E,l,m \rangle=\frac{\hbar}{\sqrt{M k}}\delta\left(E-\frac{\hbar^2 k^2 }{2M}\right) Y_l^m({\bf\hat k}),$$ where $M$ is the mass of the particle in question. Now I wonder what the generalization of this formula is. What if we have a two particle system where the particles have 3-momenta ${\bf p_1}$ and ${\bf p_2}$ respectively. What is then $$\langle {\bf p_1}, {\bf p_2} \vert E,l,m \rangle=~???$$ The simplest thing is to copy: $$\langle {\bf p_1}, {\bf p_2} \vert E,l,m \rangle=\frac{\hbar}{\sqrt{M_1 k_1}}\delta\left(E_1-\frac{\hbar^2 k_1^2 }{2M_1}\right) Y_l^m({\bf\hat k_1}) \frac{\hbar}{\sqrt{M_2 k_2}}\delta\left(E_2-\frac{\hbar^2 k_2^2 }{2M_2}\right) Y_l^m({\bf\hat k_2})$$ but I really don't know if this makes any sense?


title: Quantum Entanglement Versus Inflation in the Early Universe?

text: **Quantum entanglement** is one of the most fascinating and mysterious phenomena in nature. It needs no interactions, or any sort of exchange for it to take place. It is possible, not against any rules of physics as far as we know, that all matter that was created in the early universe was in an entangled state. **The question is:** Is it possible to explain the uniformity and isotropy of matter in the universe, by means of quantum entanglement in the early 'days' of universe? If that could be possible, would it mean that there would be no need for the inflationary model any more? If this problem has been researched in detail, any references posted will be appreciated.


title: Electric Potential at nano scale

text: I’ve a got a question; and I am hopeful that you can provide any information or direct me to a better resource. I'm not a physicist; so please correct me if I'm wrong. Scanning Probe Microscopy (SPM) can be used to probe and map the surface of electrical potential distribution. For example, let’s say that a Scanning Kelvin probe microscopy (SKPM) was used in some experiment to provide quantitative values of surface potential by determining the potential difference between the conductive probe tip and the sample surface (say NW assemblies). I was wondering whether there is any physical formula to model/describe the behavior of the electrical potential distribution at nano/atomic scale. Also, do you think Maxwell's equations holds? I appreciate any input.


title: Axiomatic structure behind Dirac's formulation of QM?

text: According to the paper Quantum Mechanics Beyond Hilbert Space by J.P. Antoine, several mathematical structures have been devised to make mathematical sense of Dirac's formulation of quantum mechanics (specifically, QM with operators having continuous spectra). I was wondering if there was a general set of axioms that could be specified that we would want any such structure to satisfy for it to be a sufficient mathematical image of quantum mechanics.


title: Physics history book with some math

text: I am looking for a book explaining physics from, say, Galileo and Newton till now; the book should be written using some math, similarly to my physics books when I was at the university (Halliday Resnik). I am searching a book that take care of explaining the true "evolution" of reasoning, discoveries, experiment to prove or disprove something. I do not like a book that gives definitions without explaining what had happened. I also like to find, on that book, explanations on how actual measurements are taken (how can we measure the speed of light, the mass of celestial bodies, the attributes of electron, and how were the measurements taken in the past - how was possible to measure the speed of light in 1700-1800?). I am not very interested in recent theories like strings, branes.... so it is OK if these are not discussed. I like to find a lot of discussions on concept that are related, like for instance the implications on space dimensionalities, ether non-existence, strange facts, missing explanations.


title: How does the speed of an incoming pitch affect the speed of a baseball after it's hit?

text: Which will go further if a batter manages to hit it with a baseball bat: a baseball thrown to the batter at 90 miles per hour or one thrown at 60 miles per hour?


title: What does "clockwise" mean, exactly?

text: I am in the middle of a discussion with a friend about the meaning of the term "clockwise". Wikipedia indicates that a clockwise rotation goes as **top-right-down-left**. However, my friend argues that "the clock is facing opposite to you. So if you rotate top-right-down-left facing the side the clock is facing then it is anti-clockwise." Our question, therefore, is > does "clockwise" mean **top-right-down-left** or **left-down-right-top**?


title: Irreducible tensors concept

text: This might be a little naive question, but I am having difficulty grasping the concept of irreducible tensors. Particularly, why do we decompose tensors into symmetric and anti-symmetric parts? I have not found a justification for this in my readings and would be happy to gain some intuition here.


title: Erratic light spot in optical projection

text: I once had an old microscope, that included a projection screen that could be mounted instead the eyepiece. It showed a quite decent palm-sized image. Now I have a new microscope, and removed the eyepiece to mount a single lens reflex camera. However, the image projected on the ground glass, and also on the imaging sensor , shows a brightened fluff around the image center. By moving the microscope in respect to the camera, the spot can be moved a little, but would always stay inside the field of view. What is the reason of this effect ?


title: Why is there no (time derivative of charge density) in the $B$ field in Jefimenko's equations?

text: I was going through Griffiths chapter on potentials and fields just to brush up on a few old things. He gets to Jefimenko's equations by this general path:   1. Maxwell's equations   2. Introduce scalar and vector potentials   3. Reduce MW's equations to the two equations with only potentials (no fields)   4. Choose the Lorentz gauge, which frames the two equations as 4-D Poisson equations: $$\Box^2V = -\rho/\epsilon, \quad \Box^2 \vec A = -\mu\vec J$$   5. We already know the solutions to the static case ($\dot{\rho},\dot{\vec J} = 0$ means $\Box^2$ becomes $\nabla^2$)   6. He says and then briefly proves that to get the non-static answers, we just use the retarded time $t_r = t - |r - r'|/c$ in the integrals for the static case   7. So now we have the potentials for a non-static source, and we just plug them into $\vec E = -\nabla V - \frac{\partial \vec A}{\partial t}$ and $\vec B = \nabla \times \vec A$ which apparently give us Jefimenko's equations (say $R = |r - r'|$, $\hat R$ is the unit vector in the direction of $R$, and $J$ and all $r$'s are vectors): $$E(r, t) = (1/4\pi\epsilon)\int[\frac{\rho(r',t_r)}{R^2}\hat{R} + \frac{\dot{\rho}(r',t_r)}{cR}\hat{R} - \frac{\dot{J} (r',t_r)}{c^2R}]d\tau '$$ $$B(r, t) = (\mu/4\pi)\int[\frac{J(r',t_r)}{R^2} + \frac{\dot{J}(r',t_r)}{cR}] \times \hat{R} \ d\tau '$$ So it all makes sense to me, the derivation anyway. But is there a simple intuitive reason for why the $E$ equation has a $\dot J$ term but the $B$ term has no $\dot \rho$ term? I'd suspect it has something to do with the "asymmetry" in Maxwell's equations with respect to $E$ and $B$ (which I also don't really understand), but that doesn't answer much. Also, what are some concrete examples of the difference between $\dot \rho$ and $\dot J$? What I mean is, it seems like they must usually be very linked, because if you have a changing amount of charge $\dot \rho$ at some point, that implies that there is a changing amount of current $\dot J$ at the point also, because the charge at that spot has to go somewhere (so $\dot J \neq 0$) for $\rho$ to change. I guess I could think of some mythical (to me anyway, because I don't know if this is possible) chemical reaction where charge at a point "disappears" and "reappears" without having to actually move spatially. I have one more small point of confusion about Jefimenko's equations: Griffiths says "In practice Jefimenko's equations are of limited utility, since it's typically easier to calculate the retarded potentials and differentiate them, rather than going directly to the fields." Isn't that exactly what he did in the process I described?


title: Invariant spacetime - distance - Circular Motion

text: I understand that the closer something travels to the speed of light, that time will stretch by a factor, and distance will compress by the same factor. My question is, if something travels in a circle, close to the speed of light, what does the distance of the journey look like to them? They measure that the trip took them 10 minutes. And an outside observer says that the journey took 20 minutes, and the outside observer measured that they did, for example, 1000 laps of a circle circumference 1000 km. So if the plane had a distance trip counter, what would it read? And if they were looking out of the window, would the circle still look like it had a circumference of 1000km?


title: A question about the relativity of time

text: > **Possible Duplicate:**   >  Invariant spacetime - distance - Circular Motion I understand that the closer something travels to the speed of light, that time will stretch by a factor, and distance will compress by the same factor. My question is, if something travels in a circle, close to the speed of light, what does the distance of the journey look like to them? They measure that the trip took them 10 minutes. And an outside observer says that the journey took 20 minutes, and the outside observer measured that they did, for example, 1000 laps of a circle circumference 1000 km. So if the plane had a distance trip counter, what would it read? And if they were looking out of the window, would the circle still look like it had a circumference of 1000km?


title: Paradox of the Relativistic Record Player

text: > **Possible Duplicate:**   >  Invariant spacetime - distance - Circular Motion This is a question that I thought up a few years ago when I was taking mechanics. I asked the professor but didn't really get a straight answer. Imagine a record spinning at relativistic speeds. In the lab frame, the circumference of the record should decrease according to the Lorentz contraction. However the radius of the record should remain fixed since it is orthogonal to the direction of motion. So does the shape of the record appear distorted since the circumference is smaller but the radius is the same? What would it look like? I have a feeling the answer is obvious and/or well known since it would seem to be similar to the situation of relativistic particles traveling around a circular accelerator but I can't think of how it can be solved.


title: What are the relativistic effects of expanding spacetime?

text: This is a question I've been mulling over for a while and I'm hoping someone here can point me in the right direction. Sorry if it's a bit of a novice question. For the record, I don't fully know GR, but don't let that stop you from using it in the answer. Since the universe is expanding - that is, the spacetime metric is expanding by way of a near-exponentially increasing scale factor - we can say that the distance between any two non-bound objects is increasing over time. Herein lays my dilemma; if there were two objects separated by a large distance that had no relative velocities initially, after a long time, the effects of expansion would cause them to have large apparent velocities away from each other. Given that there hasn't been any acceleration to cause these velocities, are there still relativistic effects in play? That is, is there time dilation between the two frames? Furthermore, given long enough time, the rate of increasing distance between the two objects could place them outside of their visible horizon (ie they are travelling away from each other at superluminal velocities). Since there was still no acceleration to achieve this feat, what can one say about the relativistic effects in this case? At first I thought this was an easy question. I thought of course there would be relativistic effects and when the objects go superluminal, the visible horizon is there to ensure there can never be causal contact and thus preserve physics. But then I thought what if spacetime stopped expanding abruptly (seems crazy but as far as I know, nothing makes this completely impossible)? Since there was no initial relative velocities, wouldn't the two objects return to being in the same inertial frame? And seeing as none of them experienced any sort of acceleration, how then could we describe their two final states? By which I mean, if we were to assume there were relativistic effects during transit, how would we overcome such simple paradoxes like the twin paradox, or other relevant ones? At this point, I'm stumped. I even attended a lecture by Miguel Alcubierre since he would have had to consider these types of effects in his design... No help. Equations are great to illustrate a point, but I'm really going to need a conceptual answer as well to fully understand this.


title: Einstein's mass-energy relation

text: Suppose we have 1 kg of wood and 1 kg of uranium and if we need to find out how much energy would each of the substance give, we'd have to use Einstein's mass-energy relation as follows: In the case of wood, $E_{wood} = 1 × (3×10^8)^2 = 9 × 10^{16} J$ In case of Uranium, $E_{uranium} = 1 × (3×10^8)^2 = 9 × 10^{16} J$ _Question:_ According to the equation, both give the same amount of energy. But in reality, 1 kg of uranium will give a lot of energy when compared to 1 kg of wood. What have I missed? Any explanation regarding the $E=mc^2$ _problem_ would be helpful...


title: Would a spin-2 particle necessarily have to be a graviton?

text: I'm reading often that a possible reason to explain why the Nobel committee is coping out from making the physics Nobel related to the higgs could be among other things the fact that the spin of the new particle has not yet been definitively determined, it could still be 0 or 2. This makes me wonder if the spin would (very very surprisingly!) finally be discovered to be 2, this then necessarily would mean that the particle has to be a graviton? Or could there hypothetically be other spin-2 particles? If not, why not and if there indeed exist other possibilities what would they be?


title: What is meant by the term "mean rest frame"?

text: I found this part in the book of Bernard Schutz on GR. (Sec 12.2: General Relativistic Cosmological Models). > Robertson-Walker metrics: We shall adopt the following assumptions about the > universe: (i) spacetime can be sliced into hypersurfaces of constant time > which are perfectly homogeneous and isotropic; and (ii) the mean rest frame > of the galaxies agrees with the definition of simultaneity. Can someone kindly explain the second assumption, in particular what is a "mean rest frame" for a galaxy.


title: positive energies

text: if the potential is bounded below $ V(x)=V(-x) \ge a $ for some real number a and we can be sure that as $ x\rightarrow \infty $ we know that $ V(x) \ge 0 $ then does it mean that the energies for our one dimensional system will be positive ?? my idea WKB quntization $ N(E)= 2\int_{0}^{a} \sqrt {E_{n}-V(x)} $ here $ V(a)=E $ is a turnign point so if the potential is positive for big 'x' then the energies should be also positive otherwise the epxresion inside the integral would be complex am i right ?