Finding force exerted in an Inelastic Collision

I did a lab today in Physics in which we launched ball from a spring loaded cannon directly into a pendulum that captured the ball, held it, and swung upwards with it (representing a totally inelastic collision). One question in particular has confused me: > If the collision between the projectile and pendulum had lasted 1 > millisecond, what would the average force have been which the projectile > exerted on the pendulum for the long-range case? My attempt at a solution is as follows: From all the searching I've been doing online, I've found the equation $F = {{p_f}-{p_i}\over {t}}$. I know $p_f$, $p_i$, and I'm given t. Is my understanding right? Can I go right ahead and crunch these numbers, or do I have an incorrect equation?


Extended Born relativity, Nambu 3-form and ternary (n-ary) symmetry

**Background:** Classical Mechanics is based on the Poincare-Cartan two-form $$\omega_2=dx\wedge dp$$ where $p=\dot{x}$. Quantum mechanics is secretly a subtle modification of this. By the other hand, the so-called Born-reciprocal relativity is based on the "phase-space"-like metric $$ds^2=dx^2-c^2dt^2+Adp^2-BdE^2$$ and its full space-time+phase-space extension: $$ds^2=dX^2+dP^2=dx^\mu dx_\mu+\dfrac{1}{\lambda^2}dp^\nu dp_\nu$$ where $$P=\dot{X}$$ Note: particle-wave duality is something like $ x^\mu=\dfrac{h}{p_\mu}$. In Born's reciprocal relativity you have the invariance group which is the _intersection_ of SO (4 +4) and the ordinary symplectic group Sp (4), related to the invariance under the symplectic transformations leaving the Poincaré- Cartan two-form invariant. The intersection of SO(8) and Sp(4) gives you, essentially, the unitary group U (4), or some "cousin" closely related to the metaplectic group. We can try to guess an extension of Born's reciprocal relativity based on higher accelerations as an interesting academical exercise (at least it is for me). In order to do it, you have to find a symmetry which leaves spacetime+phasespace invariant, the force-momentum-space-time extended Born space-time+phase-space interval $ds^2=dx^2+dp^2+df^2$ with $p=\dot{x}$, $ f=\dot{p}$ in this set up. Note that is is the most simple extension, but I am also interested in the problem to enlarge it to extra derivatives, like Tug, Yank,...and n-order derivatives of position. Let me continue. This last metric looks invariant under an orthogonal group SO (4+4+4) = SO (12) group (you can forget about signatures at this moment). One also needs to have an invariant triple wedge product three-form $$\omega_3=d X\wedge dP \wedge d F$$ something tha seems to be connected with a Nambu structure and where $P=\dot{X}$ and $F=\dot{P}$ and with invariance under the (ternary) 3-ary "symplectic" transformations leaving the above 3-form invariant. **My Question(s):** I am trying to discover some (likely nontrivial) Born- reciprocal like generalized transformations for the case of "higher-order" Born-reciprocal like relativities (I am interested in that topic for more than one reason I can not tell you here). I do know what the phase-space Born- reciprocal invariance group transformations ARE (you can see them,e.g., in this nice thesis BornRelthesis) in the case of reciprocal relativity (as I told you above). So, my question, which comes from the original author of the extended Born-phase space relativity, **Carlos Castro Perelman in** this paper, and references therein, is a natural question in the context of higher- order Finsler-like extensions of Special Relativity, and it eventually would include the important issue of curved (generalized) relativistic phase-space- time. After the above preliminary stuff, the issue is: > What is the intersection of the group SO (12) with the _ternary_ group which > leaves invariant the triple-wedge product > > $$\omega_3=d X\wedge dP \wedge d F$$ More generally, I am in fact interested in the next problem. So the extra or bonus question is: what is the (n-ary?) group structure leaving invariant the (n+1)-form $$ \omega_{n+1}=dx\wedge dp\wedge d\dot{p}\wedge\cdots \wedge dp^{(n-1)}$$ where there we include up to (n-1) derivatives of momentum in the exterior product or equivalently $$ \omega_{n+1}=dx\wedge d\dot{x}\wedge d\ddot{x}\wedge\cdots \wedge dx^{(n)}$$ contains up to the n-th derivative of the position. In this case the higher- order metric would be: $$ds^2=dX^2+dP^2+dF^2+\ldots+dP^{(n-1)}=dX^2+d\dot{X}^2+d\ddot{X}^2+\ldots+dX^{(n)2}$$ This metric is invariant under SO(4(n+1)) symmetry (if we work in 4D spacetime), but what is the symmetry group or invariance of the above (n+1)-form and whose intersection with the SO(4(n+1)) group gives us the higher-order generalization of the U(4)/metaplectic invariance group of Born's reciprocal relativity in phase-space? This knowledge should allow me (us) to find the analogue of the (nontrivial) Lorentz transformations which mix the $X,\dot{X}=P,\ddot{X}=\dot{P}=F,\ldots$ coordinates in this enlarged Born relativity theory. **Remark:** In the case we include no derivatives in the "generalized phase space" of position (or we don't include any momentum coordinate in the metric) we get the usual SR/GR metric. When n=1, we get phase space relativity. When n=2, we would obtain the first of a higher-order space-time-momentum-force generalized Born relativity. I am interested in that because one of my main research topics are generalized/enlarged/enhacend/extended theories of relativity. I firmly believe we have not exhausted the power of the relativity principle in every possible direction. I do know what the transformation are in the case where one only has X and P. I need help to find and work out myself the nontrivial transformations mixing X,P and higher order derivatives...The higher-order extension of Lorentz-Born symmetry/transformation group of special/reciprocal relativity.


In which direction is the acceleration directed in a non uniform circular motion?

Acceleration is directed towards the center of the circle in a uniform circular motion. Is it same for the non-uniform circular motion?


Ex 0.2.1 in Sachs and Wu's textbook

In the next attachements are: 1\. Exercise 0.2.5 which I want help with.   1. Proposition 0.2.1 and its proof. Now, basically a few things are changed in the theorem, I don't think I can use here the definition of s(t) in the proof of prop0.2.1 cause its s(t)=0, I don't think I can use this trick here. Other thoughts that I had, obviously if I plug m=0 into prop0.2.1 I get that I should have: $$\frac{d\gamma^1}{du}=\pm \frac{d\gamma^2}{du}$$, and $$\frac{d\gamma^2}{du}=a$$. My question is how do I satisfy condition b in the theorem, I guess this x should be $$\pm Id +constant$$ ![1](http://i.stack.imgur.com/uNJZS.png) ![2](http://i.stack.imgur.com/Gftvm.png) ![3](http://i.stack.imgur.com/Eo0PX.png)


Angular momentum of the electric field of a point-like electric charge and the magnetic field of a monopole

I am currently reading "Magnetic Monopoles" of Ya. Shnir. My problem is I can not retrieve a result the author provides in the first chapter of the first part. In this chapter, he studies the non-relativistic scattering of an electric charge on a magnetic one. The author writes [p.5, near eq. (1.13)]: > ... the appearance of an additional term in the definition of the angular > momentum $(1.11)$ originates from a non-trivial field contribution. Indeed, > since a static monopole is placed at the origin, its magnetic field is given > by $(1.1)$. Then the classical angular momentum of the electric field of a > point-like electric charge, whose position is defined by its radius vector > $\mathbf{r}$, and the magnetic field of a monopole is a volume integral > involving the Poynting vector > > \begin{align} \tilde{\mathbf{L}}_{eg} &= \dfrac{1}{4\pi}\int \mathbf{r'} > \times \left [ \mathbf{E} \times \mathbf{B}\right] d^3r'\tag{L.1}\\\& = - > \dfrac{g}{4\pi} \int d^3r' \left( \mathbf{\nabla}'\cdot \mathbf{E}\right) > \hat{\bf r}' \tag{L.2}\\\ &= -eg\hat{\bf r} \tag{L.3} \end{align} > > where we perform the integration by parts, take into account that the fields > vanish asymptotically and invoke the Maxwell equation > > \begin{equation}\left(\mathbf{\nabla}' . \mathbf{E} \right) = 4 \pi e > \delta^{(3)}\left( \mathbf{r} - \mathbf{r}'\right)\end{equation} ... The magnetic field is $\mathbf{B} = \dfrac{g}{r^3} \mathbf{r} \tag{1.1}$ The generalised angular momentum is $\mathbf{L} = \mathbf{r} \times m\mathbf{v} - eg \hat{\bf r} \tag{1.11}$ The author gives how he got $(L.2)$ from $(L.1)$ but I do not know how to do? Have you any idea?


Why does your reflection stay the same size when you move further away from the mirror?

This was an experiment I saw in my son's workbook. It said to mark out the top of your forehead and the bottom of your chin on a mirror using a whiteboard marker. Then slowly move backwards, and investigate what happens to the size of the reflection subjective to the two marks made. It actually got me quite flabbergasted. I always thought the reflection would get smaller as you moved away from the mirror. Why is this?


Max speed as a function of engine power

My question started out as finding the maximum speed of a go kart, taking into account only the drag forces as force opposing the motor. I've done some investigation to find: $$ F_{drag}=\; \frac{\rho v^{2}\ C_{x} A_{f}}{2} $$ Where $C$ = drag coefficient, and $\rho$ is the density of air. Further, with $ F_{motor}=\dfrac{P_{motor}}{v} $ and the condition that at maximum speed, the acceleration will be 0, net force will be 0 as well, so $F_{motor}=F_{drag}$. Combining equations, $F_{drag}=\dfrac{P_{motor}}{v}$, so $\dfrac{P_{motor}}{v}=\; \dfrac{\rho v^{2} C A_{f}}{2}$ Simplifying gives $P_{motor}=\; \dfrac{\rho\cdot v^{3}C\cdot A_{f}}{2}$ Solving for $v$ yields: $$v=\; \sqrt[3]{\dfrac{2P_{motor}}{\rho C A_{f}}}$$ Converting to units of miles per hour for $v$, and horsepower for $P_{motor}$: $$v=\; 2.2 \sqrt[3]{\dfrac{HP_{motor} \cdot 745\cdot 2}{\rho\mbox{C}A_{f}}}$$ From this pdf, I've found values of $\rho = 1.2 \frac{kg}{m^{3}}$, $C=.8$, and $A_{f}=.57$. So! Questions:   1. Are my equations correct? I'm particularly concerned about the third root portion, given that drag is a quadratic. Doesn't connect in my mind, so perhaps my substitutions are faulty?   2. Are the values I have for those constants for ideal values, (i.e. perfectly enginneered kart, very unlike what I'd be able to construct myself). If so, by what general percent will my values be different?   3. Finally, is it unrealistic to call all forces other than drag insignificant? If so, by what general percent will my values for speed be too high? For reference, this calculation has a 5 hp engine at a max speed of 52 mph, 15 hp at 75 mph, and 35 hp at 100 mph.


Why lunar day lasts exactly one month?

Therefore we don't see dark side of the moon. Is the core of the Moon closer to the Earth than the center of the moon? Or what is the reason? ![Lunar motion](http://i.stack.imgur.com/xZ3f6.jpg)


How to get funding for research on something that can revolutionize the quantum world!

I think this can revolutionize the quantum world! Any ideas on how to impress physicists to get a full fledged funding for research?


Moduli potential in Type IIB String Theory

In the book _String Theory and M-Theory_ by K. Becker, M. Becker and J.H. Schwarz:   1. Why is the potential for moduli given by eq (10.168): $$\tag{10.168 }V(T,K) ~=~ \frac1{4\mathcal{V}^3} \Big( \int_{CY_4} F \wedge \star F - \frac16 \chi T_{M2} \Big)?$$   Maybe the answer of this question is trivial, but I cannot see how $T$ and $K$ enter this picture and get $V(T,K)$ as their potential.   2. How can one arrives at eq (10.181): $$\tag{10.181} V~=~e^{\mathcal{K}} \Big( G^{a \bar{b}} \mathcal{D}_a W \mathcal{D}_\bar{b} \bar{W} - 3|W|^2 \Big)?$$


Vacuum Expectation Value and the Minima of the Potential

Often times in quantum field theory, you will hear people using the term "vacuum expectation value" when referring to the minimum of the potential $V(\phi )$ in the Lagrangian (I'm pretty sure every source I've seen that explains the Higgs mechanism uses this terminology). However, a priori, it would seem that the term "vacuum expectation value" (of a field $\phi$) should refer to $\langle 0|\phi |0\rangle$, where $|0\rangle$ is the physical vacuum of the theory (whatever that means; see my other question). What is the proof that these two coincide?


If you flew into a black hole

Would you outlive everyone? I'm coming from the point of view that time would be experienced more slowly (although not from your point of view) the denser the gravity gets.


A dielectric table is being inserted between a plate capacitor and $\triangle U<0$ how to deduce the table is attracted to the plates?

I am practicing for an exam in my Physics $2$ course. One of a previews exam questions described a plate capacitor and asked to calculate the initial energy $U_{0}$, then a dielectric table was inserted between the plates and I was asked to calculate the energy, $U_{1}$, at this state. I got that there was less energy in the final state than there was at the initial state, i.e $$ \triangle U=U_{1}-U_{0}<0 $$ The question asked if the table is attracted to the plates or repealed from it, and the answer claimed that since there was less energy at the final state then it means that the table is attracted to the plates. I lack intuition on this and I don't understand how the conclusion was made. I tried to think it with terms of work done by the electrical field: If I assume that the table is attracted to the plates then I think that the work is negative (since I don't have to do actual work because there is an attraction), and since there was negative work done $W<0$ I have $$U_{1}=U_{0}+W<U_{0}$$ The other case leads to $U_{1}>U_{0}$ and so I get that the first is the one that occurs. But I am not to sure about this argument since $$W=\int F\cdot dl$$ and since there is an attraction the direction of the path is the same as the direction of the force and so $W>0$. Can someone please help me understand why there is an attraction ?


A question about an identity in deriving Born-Infeld action

I have a question in David Tong's Example Sheet 4 Problem 5b, how to verify the last equation (*) on p.2? (There is a solution for example sheet 3, but seems to be no solution for example sheet 4.) > Problem 5b: > > Show that the equations of motion arising from the Born-Infeld action are > equivalent to the beta function condition for the open string,   >  $$\beta_\sigma\left(F\right)=\left( \frac{1} {1 -F^2} \right)^{\mu > \rho}\partial_\mu F_{\rho\sigma }=0 $$ **Note:** To do this, it will prove > very useful if you can ﬁrst show the following results:   >  $$∂_μ\left[\operatorname{tr} \ln(1 − F^2)\right] = −4 ∂_\rho > F_{μ\sigma}\left(\frac{F}{1-F^2}\right)^{\sigma \rho } $$ > > which requires use of the Bianchi identity for $F_{\mu \nu }$ and > > $$ \tag{*} \begin{align} \partial _\mu \left( \frac{ F}{1-F^2} > \right)^{\mu\nu} &= \left( \frac{ F}{1-F^2} \right)^{\mu\rho} \partial_\mu > F_{\rho\sigma} \left( \frac{ F}{1-F^2} \right)^{\sigma \nu} \\\ &\qquad+ > \left( \frac{1} {1 -F^2} \right)^{\mu \rho} \partial_\mu > F_{\rho\sigma}\left( \frac{ 1 }{1-F^2} \right)^{\sigma \nu} \end{align}$$ In addition, as given in question 5a > $$F_{\mu\nu} = \partial_{\mu} A_{\nu} - \partial_{\nu} A_{\mu} $$ My attempt to prove the problem: LHS $$\partial_{\mu} \left( \frac{ F}{1-F^2} \right)^{\mu\nu} = \partial_{\mu} \left[ F^{\mu}_{\alpha} \left( \frac{1}{1-F^2} \right)^{\alpha \nu} \right] = \left( \partial_{\mu} F^{\mu}_{\alpha} \right) \left( \frac{1}{1 -F^2} \right)^{\alpha \nu} + F^{\mu}_{\alpha} \partial_{\mu} \left( \frac{1}{1 -F^2} \right)^{\alpha \nu} \tag{1} $$ Using the formula in matrix cookbook for the derivative of inverse matrix, Eq. (53) Eq. (1) becomes $$\left( \partial_{\mu} F^{\mu}_{\alpha} \right) \left( \frac{1}{1 -F^2} \right)^{\alpha \nu} + 2 F^{\mu}_{\alpha} \left[ \frac{1}{1 -F^2} \left( \partial_{\mu} F \right) F \frac{1}{1-F^2} \right]^{\alpha \nu} $$   $$= \left( \partial_{\mu} F^{\mu}_{\alpha} \right) \left( \frac{1}{1 -F^2} \right)^{\alpha \nu} + 2 \left( \frac{F}{1 -F^2} \right)^{\mu \rho} \left( \partial_{\mu} F \right)_{\rho\sigma} \left(\frac{F}{1-F^2}\right)^{\sigma \nu} \tag{2} $$ The second term in Eq.(2) cancels the second term in the RHS in the problem sheet equation. We then need to show $$ \left( \partial_{\mu} F^{\mu}_{\alpha} \right) \left( \frac{1}{1 -F^2} \right)^{\alpha \nu} + \left( \frac{F}{1 -F^2} \right)^{\mu \rho} \left( \partial_{\mu} F \right)_{\rho\sigma} \left(\frac{F}{1-F^2}\right)^{\sigma \nu} - \left( \frac{1}{1 -F^2} \right)^{\mu \rho} \left( \partial_{\mu} F \right)_{\rho\sigma} \left(\frac{1}{1-F^2}\right)^{\sigma \nu} =0 \tag{3} $$ then I didn't find a way to show Eq. (3) hold. I tried to combine the second and third term, and rearrange them, but didn't got a simple expression.


A question about the constraints in BRST-Fock theories

In _BRST Symmetry in the Classical and Quantum Theories of Gauge Systems_ , Henneaux says the Fock representation is not applicable to an odd number of constraints. Then he goes on to say that the Kugo-Ojima quartet requires the constraints to be in pairs. For BRST theories, when are they not in pairs?


Looking for solutions to problems in Feynman and Hibbs path integral and QM text

I've just started reading Feynman and Hibbs path integrals and Quantum mechanics after a decade hiatus from my undergraduate math degree (including a few semesters of physics for engineers). It would be tremendously helpful to see the step by step solutions to some of the first set of problems (p27-28, #2.1 and on) As many as folks are willing to post! Or if anyone knows of the solutions being available on-line that would also be helpful. The specific concept question is finding the extremum of functional that is the integral of the lagrangian of a system (classical action). The easiest example is 2.1, show that for L=m/2 (dx/dt)^2, the extremum is m/2*(Xb- xa)^2/(tb-ta). I have tried manipulating the integral by replacing 'partial of L with respect to x' with 'd/dt (partial L with respect to dx/dt) but haven't gotten there. I know it's the easiest problem but I think if I see an example, I will be able to apply what I learn to the harder subsequent problems I hope that's enough to come off "hold" Thanks!


Capacitors in series: Why is the equivalent charge the same as the individual charges

Consider two parallel-plate capacitors $C_1, C_2$ in series. For the "equivalent" circuit, clearly $$ Q_{equiv} = C_{equiv}V$$ should hold, where V is the total voltage drop between input and output. It is also obvious that $$ Q_1 = Q_2 $$ where $Q_1$ and $Q_2$ are the charges (plus and minus on opposite sides) accumulated on each of the two sets of plates. Obviously in general, $$ V_1 \neq V_2 \qquad \text{}$$ However, just because we have two capacitors with equal charge inside the circuit why is it true that for the equivalent circuit $$Q_{equiv} = Q_1 = Q_2$$ I see no obvious reason why this should be true, and this is _assumed_ , though not explained in the discusssion below: http://farside.ph.utexas.edu/teaching/302l/lectures/node46.html


Reconciling topological insulators and topological order

We make an important distinction between the topological insulators (which are essentially uncorrelated band insulators, "with a twist") and topological order (which covers a variety of exotic properties in certain quantum many- body ground states). The topological insulators are clearly "topological" in the sense of the connectedness of the single particle Hilbert space for one electron; however they are not "robust" in the same way as topologically ordered matter. My question is this: Topological order is certainly the more general and intriguing situation, but the notion of "topology" seems actually less explicit than in the topological insulators. Is there an easy way to reconcile this? Perhaps a starting point might be, can we imagine a "topological insulator in Fock space"? Would such a beast have "long range entanglement" and "topological order"? **Edit:** While this has received very nice answers, I should maybe clarify what I'm looking for a bit; I'm aware of the "standard definitions" of (symmetry protected) topological insulators and topological order and why they are very different phenomena. However, if I'm talking to nonexperts, I can describe topological insulators as, more or less, "Berry phases can give rise to a nontrivial 'band geometry,' and analogous to Gauss-Bonnet there is a nice quantity calculable from this that characterizes instead the 'band topology' and this quantity is also physically measurable" and they seem quite happy with this. On the other hand, while the connection to something like Gauss-Bonnet might be clear for topological order in "TQFTs" or in the ground state degeneracy, these seem a bit formal. I think my favorite answer is the adiabatic continuity (or lack thereof) that Everett pointed out, but now that I'm thinking about it perhaps what I should have asked for is -- What are the _geometric_ properties of states with topological order from which we could deduce the topological order with some kind of Chern number (but without starting from a Chern-Simons field theory and putting in the right one by hand ;) ). Is there anything like this?


Redshift Mechanism

How does the stretching/expansion of space cause redshift in light from distant stars? What is the mechanism that causes the electromagnetic crests to be further apart?


What is the relationship between the Higgs field and quarks?

I have some difficulty considering the relative size of each and the meaning behind the shape of Higgs boson. I ask relating to the _structures_ of both the Higgs field and quarks. How is it that the structure of a Higgs boson flows into that of, for instance, a bottom-antibottom quark pair? Essentially I am asking (or at least think I am asking): If the interactions for the field to exist occurred at some point in the universe's past, the particle is expressing it's shape in relation to the field, etc, etc.. Does this mean {when viewing some of the type of symmetries seen in readouts of the possible Higgs boson decay} quarks themselves are further expressions of the same field's shape or instead some manner of deformation? ![Now well known shape of a Higgs Boson, computer generated from Wikipedia](http://upload.wikimedia.org/wikipedia/commons/thumb/1/1c/CMS_Higgs- event.jpg/650px-CMS_Higgs-event.jpg) This now fairly well known image from Wikipedia is a computer generated Higgs boson demonstrating simulated decay trajectories. This has often given me some considerations and can hopefully serve to slightly illustrate the structures I'm inquiring. (Knowing this is neither the boson or the quarks themselves)


Distinguishing between an entangled and non-entangled state (mainly $S(H_A) \otimes S(H_B)$ vs. $S(H_A \otimes H_B)$)

Say I have two quantum systems $A$ and $B$ I can look at the joint (composite) system $AB$ which is given by $H_{AB} \in H_A \otimes H_B$ Measuring a subsystem with respect to a collection of measurement matrices $\textbf{M} = \\{M_i\\}_{i \in I} \in Meas_{I}(H_A)$ acts as measuring $AB$ with respect to $\textbf{M} \otimes \mathbb{I}_B = \\{M_i \otimes \mathbb{I}_B\\}_{i \in I}$ Q: Can I do this with entangled states? I know that making a measurement causes the entangled state of say 2 qubits to decompose in to two states. If I understand correctly, they are not entangled anymore. Now because of this we can seperate the states in to a sum of the product states $|AB> = \sum \alpha_j|j> \otimes |\psi_j>$ over all basis states $|j> \in S(H_A)$. What this says to me is that we can some how distinguish between these qubit states (hence they are not entangled anymore). Do I have this right? This leads me in to the difference between $S(H_A) \otimes S(H_B)$ vs. $S(H_A \otimes H_B)$. Now the first case is not entangled and we have the "product states" of two wavefunctions/state vectors , but in the second case we have some sort of combination (composite?) of states? I would guess this is when the states are entangled and after measurement they decompose in to say $|\psi> \otimes |\phi> \in S(H_A) \otimes S(H_B)$. Now if I look at an entangled 2-qubit state that I know is entangled $|\Phi> = |\Phi^+> = \frac{1}{\sqrt{2}}(|0>|0> \+ |1>|1>) \in H_A \otimes H_B$ (as written out in some notes) I see that we can have an entangled state embedded somehow in the tensor of two state vectors in Hilbert space. I don't know if this is supposed to be $S(H_A \otimes H_B)$ or $S(H_A) \otimes S(H_B)$ or something else completely. Edit: I removed some tensor math that was incorrect. I had initially thought that if some mixture of states could be decomposed and have just a $\otimes$ and nothing such as an addition or subtraction operator it. I know that is very rudimentary but from my basic understanding of product states: http://en.wikipedia.org/wiki/Product_state I see that if a probability density can be written as the tensor product of two different probability densities, then we have a non-quantum correlation, although in the wiki article above, this is neither quantum or classical in nature. I however also see that a mixed state such as $\rho_{AB}=\frac{1}{2}(|0_A0_B⟩⟨0_A0_B|+|1_A1_B⟩⟨1_A1_B|)$ only has classical correlations. Furthermore I have found that Two states $\rho, \sigma$ are called $\textbf{perfectly distinguishable}$ if there exists a measurement $M \in Meas(H) \text{ with } I = {0,1,...}$ such that $p_0(M,\rho) = 1 = p_1(M,\sigma)$. Now from my reading, in the case of a von Neumann measurement $M = \\{M_i\\}, p_i$ simplifies to $p_i = tr(M_i \rho)$ and in the case of a complete von Neumann measurement $M = \\{ |i \rangle \langle i | \\}$, $p_i$ simplifies to $p_i = tr(|i \rangle \langle i | \rho) = \langle i | \rho |i \rangle$ So basically, I just have to take the traces of the respective measurements on probability densities, if they are equal (hence perfectly distinguishable) then the system of, say qubits, are not entangled, otherwise they are? Is this correct? Thank you, Brian


Does the nonlocality of the preferred basis mean QM is nonlocal?

Take the Mach-Zehnder interferometer as an example. A photon passes through a beam splitter, is reflected off mirrors, and interferes with itself at another half-silvered mirror. No measurements or disturbances are made in between. The preferred basis of the photon is given by which path it takes after passing the second beam splitter. The problem is, this preferred basis is nonlocal in between. Is there a problem with locality here, or is silly ol' me just plain confused?


Role of unit vectors in cylindrical coordinates

I know how the unit vectors are defined in cylindrical coords. If I have a point P, how do I express it as a combination of the unit vectors uρ, uφ and uz. In the case of Cartesian coordinates this combination is linear. But what about cylindrical coords? Does such a combination exist for them? And, BTW, could you suggest a simple Physics problem where the use of cylindrical coordinates is convenient or, in general, the reason to choose them?


In QFT, why do fermions have to anticommute in order to insure causality?

I have seen this question and I believe I understand the answer to it. However, AFAIK, only for bosons the causality condition is a vanishing commutator. For fermions we expect the _anticommutator_ $[\phi,\phi^\dagger]_+$ to turn zero. The answer given to the question above does not seem to address this.


Proving Simple Harmonic Motion (direction of acceleration)

A particle of mass $m=0.5kg$, is attached to a spring of natural length $l=0.6m$ and modulus of elasticity $\lambda=60N$, and the setup is on a horizontal smooth table. The other end of the spring is attached to a fixed point $A$ on the table. The particle is then pulled so that the distance $AP=0.9m$, and is then released from rest. ($P$ is to the right of $A$, and the tension in the spring is $T$). If you take $x$ (displacement from the centre of oscillation) as increasing to the right, then you can prove SHM if you can get equation that is like this: $$a = -w^2$$ So $$-T =ma$$ (The $T$ is negative because $x$ increases to the right making right positive, and $T$ is to the left, and $a$ is positive because in SHM $a$ is always in the direction of $x$ increasing.) $$-\frac{\lambda x}{l} = ma$$ $$-\frac{60x}{0.6} = 0.5a$$ $$-200x = a$$ (which fits the equation at the top so its SHM.) However if you take $x$ to increase to the left: $$T = ma$$ ($T$ is positive because $x$ increases to the left making left positive and $T$ is to the left, $a$ is positive because, again, in SHM $a$ is always in the direction of $x$ increasing.) $$\frac{\lambda x}{l} = ma$$ $$\frac{60x}{0.6} = 0.5a$$ $$200x = a$$ (which doesn't fit the equation because there is no negative.) So I don't understand what I'm not getting right in the second part? how can changing the defining of the direction of $x$ increasing have such an effect? (Instead of changing the direction of $x$ increasing I could have said the particle $P$ is pushed so that the spring is compressed and $AP=0.3m$ making the $T$ act towards the right which is the same direction as $x$ increasing).


Maxwell's Equations using Differential Forms

Maxwell's Equations written with usual vector calculus are $$\nabla \cdot E=\rho/\epsilon_0 \qquad \nabla \cdot B=0$$ $$\nabla\times E=-\dfrac{\partial B}{\partial t} \qquad\nabla\times B=\mu_0j+\dfrac{1}{c^2}\dfrac{\partial E}{\partial t}$$ now, if we are to translate into differential forms we notice something: from the first two equations, it seems that $E$ and $B$ should be $2$-forms. The reason is simple: we are taking divergence, and divergence of a vector field is equivalent to the exterior derivative of a $2$-form, so this is the first point. The second two equations, though, suggests $E$ and $B$ should be $1$-forms, because we are taking curl. Thinking of integrals, the first two we integrate over surfaces, so the integrands should be $2$-forms and the second two we integrate over paths and so the integrands should be $1$-forms. In that case, how do we represent $E$ and $B$ with differential forms, if in each equation they should be a different kind of form?


The speed of light and fields

Does the speed of light apply to the speed of (waves?) in every known field, or does it only apply to the electromagnetic field?


Hilbert Space of (quantum) Gauge theory

Since quantum Gauge theory is a quantum mechanical theory, whether someone could explain how to construct and write down **the Hilbert Space of quantum Gauge theory with spin-S**. (Are there something more rich/subtle than just saying the Hilbert Space are composed by the state space of infinite many sets and infinite many modes of harmonic oscillators? - i.e. more rich/subtle than usual spin-0 scalar fields' Hilbert space?) Is the quantum Gauge theory's Hilbert Space written **in a tensor product form or not** (eg. thinking about put this gauge theory **on the lattice** )? Whether there are differences for this procedure construction of Hilbert space for these three cases: (1) spin-1 quantum Gauge theory with Abelian $U(1)$ symmetry (2) spin-1 quantum Gauge theory with non-Abelian (such as $SU(N)$) symmetry (3) spin-2 quantum Gauge theory (Gravity? or anything else) Also, whether gauge-redundancy plays any roles? Is there similar thing like Faddeev-Popov ghosts happened in the path integral formalism, when one dealing with gauge-redundancy?


How large is the information collected from an inverse femtobarn of  collisions?

I ran into this while looking at measures of humongous amounts of data. How does the information (data) collected in an inverse femtobarn exposure compare to a gigabyte of data ?


Do resistor-based fan regulators save no power at all?

I have heard that the traditional resistor-based fan speed regulators are inefficient. In fact, I have noticed that such regulators tend to get hot when the fan is set to low speed. However, does that mean that they save **_no power at all_** at low speeds, or is it just that power is saved but the amount of power saved is far less than ideal? I was expecting that since power P = V / R^2, the overall power consumption of the fan plus regulator system would be comparatively lower at lower speeds.


Relationship between Alcubierre drive space-time evolution and speed of gravity

The top rated answer to this question about the Alcubierre drive asserts, "spacetime can dynamically evolve in a way which apparently violates special relativity," but according to the Wikipedia article on the speed of gravity, changes in the gravitational field propagate at the speed of light. The second answer under this question puts the limitation in broader terms: "distortions of spacetime are as limited to travel to the speed of light as any other physical influence." These statements appear to contradict one another. My question is: Does the Alcubierre drive depend on space-time distortions propagating at superluminal speeds, and is that possible under general relativity? If Alcubierre drive distortions of space-time can propagate at superluminal speeds under GR then why can't gravitational waves (another traveling distortion of space-time) also propagate at superluminal speeds?


I'm an undergraduate mathematics student trying to understand some quantum mechanics, but I'm having a hard time understanding what is the status of the Schrödinger equation. In some places I've read that it's just a postulate. At least, that's how I interpret e.g. the following quote: > Where did we get that (equation) from? Nowhere. It is not possible to derive > it from anything you know. It came out of the mind of Schrödinger. -- > Richard Feynman (from the Wikipedia entry on the Schrödinger equation) However, some places seem to _derive_ the Schrödinger equation: just search for "derivation of Schrödinger equation" in google. This motivates the question in the title: Is the Schrödinger equation derived or postulated? If it is derived, then just how is it derived, and from what principles? If it is postulated, then it surely came out of somewhere. Something like "in these special cases it can be derived, and then we postulate it works in general". Or maybe not? Thanks in advance, and please bear with my physical ignorance.


Phase Plot for Harmonic Oscillator

This is probably gonna be a dumb question but I don't know exactly where I am making the mistake. I have been taught in highschool that simple harmonic oscillator phase plot is the $sin(\omega t)$: ![enter image description here](http://i.stack.imgur.com/v8EOE.png) However now that I am thinking about it, the phase is actually only the $\phi=2\pi f t$ part and therefore the real plot should look like this: ![enter image description here](http://i.stack.imgur.com/syt8f.png) What is the problem with the second plot? Shouldn't the second one be the phase plot? Note: Both of the plots are for the oscillator with $f=0.25$ and $t\in[0,10]$.


Physical significance of Taylor and Maclaurin series - What is the significance of defining a Maclaurin series in Mathematical Physics?

In physics, usually Taylor series is used to express a quantity which keep changes with coordinate. For example the potential energy of a molecule changes with coordinate, so we express the potential energy as a Taylor expansion with respect to the coordinate about the equilibrium configuration. I am confusing this with the Maclaurin series in mathematics. If the equilibrium position is zero, it is Maclaurin series. We have the freedom to fix the equilibrium position at zero. In short, what is the physical significance of Maclaurin series? or else, what is the need of defining a Maclaurin series? Any physical interests on Maclaurin series? One more question I would like to add. If a quantity is changing with respect to its coordinate and doesn't have an equilibrium at any of its configuration, then how we can express it? Taylor series can be applied only if the quantity is changing about an equilibrium configuration?


If particles can find themselves spontaneously arranged, isn't entropy actually decreasing?

Take a box of gas particles. At $t = 0$, the distribution of particles is homogeneous. There is a small probability that at $t = 1$, all particles go to the left side of the box. In this case, entropy is _decreasing_. However, it is a general principle is that entropy always _increases_. So, where is the problem please?


Does entropy really always increase (or stay the same)?

![enter image description here](http://i.stack.imgur.com/FVjLL.jpg) Consider this image. If the big (grey) molecules were all to spontaneously move to the left, and the small ones were to move to the right, there would be an increase in order. While unlikely, wouldn't this mean that entropy decreases?


Second law of Thermodynamics: Why is it only "almost" always true that entropy is non-decreasing?

Wikipedia - Second law of thermodynamics: > ...the entropy of any closed system not in thermal equilibrium **almost** > always increases. I understand that the second law of thermodynamics is based on the statistical unlikelihood of fast-moving molecules to aggregate. However, if it is only "almost always" then why is the phenomenon stated as a law? Is it because we have not yet observed the unlikely aggregation?


How light causes increase in temperature

Temperature is the measure of movements of atoms. So if something is said to have high temperature it means that its atoms are moving fast or have high KE energy. There are basically two ways heat can be travels from on object to another. 1) conduction. 2) Radiation. My question is how does radiation causes increase in KE energy of atoms, when photons only interact with electrons? Or how does electrons going up in energy state translates to atom having more KE energy.


Torque Required For a Motor to Move an Object on Wheels?

I've been attempting to calculate how much torque a motor needs to produce in order to start a stationary object on wheels moving. (The torque is being applied to the rear 2 wheels, the front 2 are on bearings.) I keep seeing Torque = Force * Radius (of the torqued wheel). I can't figure out how to calculate the Force in this equation though, so that I may find the Torque. The radius of the wheel is 3cm. The weight of the entire object (including wheels and everything else) is 5kg. I don't need an incredibly accurate result, so I haven't even been trying to factor in friction from the non-powered front wheels' bearings. I tried something with a friction coefficient and gravity, but the more I read the less I believe that my calculation was correct. Can anyone point me in the right direction?


Angular Momentum Operator

I'm looking at a review question I was given and it quite frankly has me stumped. "Using matrix representations find $L^{3}_{x},L^{3}_{y},L^{3}_{z}$ and from these show that $L_{x}, L_{y},L_{z}$ satisfy the same algebraic equations." What has me stumped is the $L^{3}_{x},L^{3}_{y},L^{3}_{z}$, I'm not even sure how it makes sense to talk of the cube of L? Any pointers in the right direction here would be incredibly helpful!


What does it mean for two objects to "touch"?

If you've ever been annoyingly poked by a geek, you might be familiar with the semi-nerdy obnoxious response of > "I'm not _actually_ touching you! The electrons in the atoms of my skin are > just getting really _close_ to yours!" Expanding on this a little bit, it seems the obnoxious geek is right. After all, consider Zeno's paradox. Every time you try to touch two objects together, you have to get them halfway there, then quarter-way, etc. In other words, there's always a infinitesimal distance in between the two objects. Atoms _don't_ "touch" each other; even the protons and neutrons in the nucleus of an atom aren't "touching" each other. So what does it mean for two objects to touch each other?   1. Are atoms that join to form a molecule "touching"? I suppose the _atoms_ are touching, because their is some overlap, but the subatomic particles are just whizzing around avoiding each other. If this is the case, should "touching" just be defined relative to some context? I.e, if I touch your hand, our hands are touching, but unless you pick up some of my DNA, the molecules in our hands aren't touching? And since the molecules aren't changing, the atoms aren't touching either?    2. Is there really no such thing as "touching"?


Is the electromagnetic force responsible for contact forces?

It is commonly stated that there are four fundamental forces, or interactions, in nature. It is natural to consider which of those is responsible for the normal force we meet in elementary physics. What is the nature of the force exerted on a body by the floor on which it rests? Of the four forces, the only one that seems relevant is the electromagnetic force, and I always assumed the repelling force in question is ultimately the EM one. The Wikipedia article on contact forces seems to agree: > Molecular and quantum physics show that the electromagnetic force is the > fundamental interaction responsible for contact forces. An alternative explanation is that the force results from Pauli's Exclusion Principle; however, reasonable as it may seem, it doesn't seem to answer the question of what _force_ is actually at play here. The exclusion principle isn't a force, or at least it's not normally described as such. I've recently come across a rather forcefully presented argument that states that the normal force is a macroscopic force that is not directly reducible to the EM force, or to any of the other forces; rather, it does indeed result from Pauli's Exclusion Principle by way of interaction of electron wave functions. The argument further claims that attempts to correct the Wikipedia article have met with the usual "source needed" objections, and that this topic is either misunderstood by or poorly covered in the standard texts. (The argument can be found here). Which way is it?


Graphene space elevator possible?

I just read this story on MIT working on industrial scale, km^2 sheet production of graphene. A quick check of Wikipedia on graphene and Wikipedia on space elevator tells me > Measurements have shown that graphene has a breaking strength 200 times > greater than steel, with a tensile strength of **130 GPa** (19,000,000 psi) and > The largest holdup to Edwards' proposed design is the technological limit of > the tether material. His calculations call for a fiber composed of epoxy- > bonded carbon nanotubes with a minimal tensile strength of **130 GPa** (19 > million psi) (including a safety factor of 2) Does this mean we may soon actually have the material for a space elevator?


Singing: Resonance body open-closed or closed-closed?

Googling yields contradictory results, so here my question: When I sing, my vocal chords vibrate, but my whole body is the resonance body, right? So I would say that when I think about standing waves etc. I must think of a resonantor that's closed (i.e. hard boundary conditions) at both ends, because my bottom end is certainly closed, and the top end has the vocal chords which are vibrating similar to woodwind instruments, so it's closed on that end as well. If I am mistaken, could someone point out the physics behind this?


Energy to move The Earth to mars' orbit?

How much energy would it take to add to the earth's orbital velocity to push the earth into mars' orbit? My thoughts: So, in 1-2 billion years, the sun will gradually get brighter. Suppose a future civilization wanted to gradually move the earth farther from the sun. And for, now, please ignore that Mars is already in Mars' orbit. let's pretend there is room there. I'm wondering what the answer looks like, in power terms, over the course of a billion years? are we talking more power than modern civilization, or if we have a billion years can we do it with just a small generator's worth of power output?


Photoelectric effect at low frequencies

Through the photoelectric effect among many others we learn that light is actually comprised of discreet quanta of energy. That's because of the energy of the emitted electrons as well as the minimum frequency of incident light that is able to start the process. A photon of frequency (energy) less than the threshold frequency is not able to free the electron from the surface. However if the required energy to free the electron was greater than this frequency by an integral value why couldn't the electron simply absorb two photons or three photons (or any required number of photons) so as to be released? In this case even with frequencies lesser than the threshold frequency there would be photoelectric effect.


Schrodinger and thermodynamics

I heard that Schrodinger pointed out that (classical/statistical) thermodynamics is impaired by logical inconsistencies and conceptual ambiguities. I am not sure why he said this and what he is talking about. Can anyone point some direction to study what he said?


Metric tensor and its inverse

1. Is it **always** allowed to represent the metric tensor $g_{\mu \nu}$ in General Relativity as a $4\times 4$ matrix?   2. If the last one is represented for example with a $4\times 4$ matrix $\mathbf{A}\in{\mathbb{R}^{4\times 4}}$ is it always true that its metric inverse $g^{\mu \nu}$ is the **ordinary** matrix inverse $\mathbf{A}^{-1}$ which one can calculate with ordinary linear algebra methods? Maybe this question may sound silly, but I've understood that in differential geometry, also simple things become complicated...


Why don't black holes have magnetic hair?

> **Possible Duplicate:**   >  What happens to an embedded magnetic field when a black hole is formed from > rotating charged dust? It is well stablished that the only hair a black hole can have is:   1. mass   2. angular momentum   3. electric charge But I can't help but wonder about the last two: a rotating charge will generate a current, that will generate a magnetic field. So, how is it even possible that the electric charge and angular momentum together will not generate a magnetic field? Since we are forbidden by the event horizon to measure any distribution of the electric charge, it means that the charge must be evenly distributed over the event horizon, and for sure we know that a rotating charged shell will generate a magnetic field What is amiss in this picture?


Why is Mendel Sachs's work not taken seriously? Or is it?

Back in college I remember coming across a few books in the physics library by Mendel Sachs. Examples are: _General Relativity and Matter_ _Quantum Mechanics and Gravity_ _Quantum Mechanics from General Relativity_ Here is something on the arXiv involving some of his work. In these books (which I note are also strangely available in most physics department libraries) he describes a program involving re-casting GR using quaternions. He does things that seem remarkable like deriving QM as a low- energy limit of GR. I don't have the GR background to unequivocally verify or reject his work, but this guy has been around for decades, and I have never found any paper or article that seriously "debunks" any of his work. It just seems like he is ignored. Are there glaring holes in his work? Is he just a complete crackpot? What is the deal?


Gauge invariant Chern-Simons Lagrangian

I have to prove the (non abelian) gauge invariance of the following lagrangian (for a certain value of $\lambda$): $$\mathcal L= -\frac14 F^{\mu\nu}_aF_{\mu\nu}^a + \frac{k}{4\pi}\epsilon^{\mu\nu\rho}\operatorname{tr}[A_{\rho}\partial_{\mu}A_{\nu} + \lambda A_{\mu}A_{\nu}A_{\rho}]$$ Is there an easier way than expliciting every component of A, going to first order gauge transformation, ... ? Cause it seems really ugly to me.


How neutrinos can be harmful?

What are the circumstances in which neutrinos can harm humans or even kill them.?


Uncertainty in momentum of an excited electron trapped in a box (1D)?

An electron is trapped in a one-dimensional well of width $0.132\,$nm. The electron is in the ninth excited state ($n=10$) state. What is the uncertainty in its momentum? The problem gives a hint to use the equation $\Delta{p_x}=\sqrt{(p_x^2)_{avg}}$ (the root mean square of the momentum of the particle). I know the wavefunction of the electron is this: $$\psi(x)=\sqrt{\frac{2}{L}}\sin\left(\frac{10\pi}{L}x\right)$$ But since the probability density function is a function of position, and not momentum, how do I calculate the RMS of the momentum from this?


What is the difference between Biot-Savart law and Ampere's law?

What is the difference between these laws? Which law is more useful? When to use Ampere's law and when to use Biot-Savart law?


Thermodynamically reversed black holes, firewalls, Casimir effect, null energy condition violations

Scott Aaronson asked a very deep question at Hawking radiation and reversibility about what happens if black hole evolution is reversed thermodynamically. Most of the commenters missed his point entirely. Of course it's overwhelmingly improbable statistically to set up the initial conditions needed to get a thermodynamically reversed black hole evolution. But that's not the point of the question at all. It is theoretically possible to have a thermodynamically reversed black hole evolution as a valid evolving state by CPT symmetry, even if it's improbable for all practical purposes. As it exists as a possible solution, we can ask questions about the properties of such a solution. In a thermodynamically probable evolution, we can have a massive object with a significant fraction of the black hole's mass fall into a black hole, and the blak hole automatically increases in mass by a huge fraction over a timescale of order of the Schwarzschild radius R. Then, it slowly evaporates away Hawking radiation with a remaining lifetime of order $R^3$. In the thermodynamically reversed version of this event, over a time period of order $R^3$, thermal radiation is fed into a black hole which increases in size very gradually as a result. There is an incredible fine-tuning of the ingoing radiation fed in with fine-tuned higher order multipartite entanglement (this is a theoretical state, not a practical engineering one) so that as a result, over the time period of order $R^3$, no Hawking radiation is _emitted_ from the black hole because of fine-tuned quantum cancellations between the contributions from the black hole, and the contribution from frequency mixing of the radiation fed into the hole. Then, after a time period of order $R^3$, suddenly over a time period of order $R$, the black hole emits as "Hawking radiation" a massive body a huge fraction of its mass, and the black hole's mass decreases accordingly. The area of its event horizon goes down by a huge fraction over a time period of order R. According to Raychaudhuri's optical equation, this requires a significant negative null energy. This is too large to come from Casimir effects, as that would require a time period of order $R^3$ to lose that much black hole mass. So, what is the origin of this huge negative null energy? Let's look at what happens just outside the black hole before the massive body is ejected. In the semiclassical approximation, the massive body must have come from a firewall ( What are cosmological "firewalls"?) at an exponentially small distance above the event horizon. This exponential factor comes from the very high boost factor and Lorentz contraction. Semiclassically, this firewall couldn't have originated from entangled Hawking pair production just outside the horizon. So, it must have originated from the fine-tuned radiation fed into the hole. What if a probe is sent during an earlier time to measure this firewall? Unfortunately, this probably can't be done. See, we are conditioning a significant part of the future state by stipulating that a huge massive body is ejected as Hawking radiation. To further specify a probe hovering just above the horizon requires in addition a significant condition on the past. Ordinarily, with no thermodynamic conspiracies, we only have past conditions and no future conditions, and so, there's no problem. However, such fine-tuned conditioning of both the past and future might actually be impossible?


Electron Positron annihilation Feynman Diagram

![](http://upload.wikimedia.org/wikipedia/commons/b/ba/Feynman_EP_Annihilation.svg) I am having some trouble understanding this fenyman diagram, it seems to indicate that the electron produces the positron, as the arrow of the positron is pointing from the electron. Additionally the arrow is in directed downwards, implying that the particle is going backwards in time? Is this diagram wrong, or does the arrow mean something else, or does the positron go back in time?!?


Why is time special?

In Special Relativity, the spacetime interval between two events is $s^2 = -(c{\Delta}t)^2+({\Delta}x)^2+({\Delta}y)^2+({\Delta}z)^2$ giving the Minkowski metric $\eta_{\mu\nu}=\text{diag}(-1, 1, 1, 1)$. What is the justification for making time have a negative coefficient, and how closely is that related to the 2nd law of thermodynamics? Sure, by letting $\eta = \text{diag}(1, 1, 1, 1)$, we get a pretty boring spacetime, and the boosts in the Poincaré group become trig instead of hyperbolic functions, but what's the physical reasoning behind this?


What is the volume of a parcel of air of a giving weight at average sea level pressure

I know this is probably some basic math once you have the figures, but my Google-fu is failing me. Say I have **5 pounds of air**. Air meaning, the typical composition of the atmosphere of Earth. And by 5 pounds I mean the weight of the mass of the air, not it's pressure or anything like that. I want to know approximately how much **volume** that amount of air would take up at a given pressure -- specifically, sea level pressure. I know that can vary -- lets just say its a typical warm summer day, or just an average. I'm not looking for exact numbers. Actually, I'd rather a subjective interpretation of the exact figure if you can offer one. Something like 'about the volume of the air in a 12x12 room', or what have you. I have no idea how big 5 pounds of air is.


Why are radio waves in the 1.43 - 2.5 Mhz range invisible?

Visible light diapason is 400 - 700 nm which is 1.43 - 2.5 Mhz. If using an antenna I would broadcast steady sinusoidal wave in this range, why the EM emitted by the antenna are not visible? Suppose the power source for antenna in a range of Kilo or Mega watts


Can a magnet spin a small machine?

If you were to build an upright machine that had a rod fitted into a ball bearing at the base and was secured to another rod at the top of the initial rod extending perpendicularly. To which you attached a magnet, and around this machine you also placed several upright rods containing magnets that would push/pull the magnet attached to the rod, would this cause the initial magnet to spin the machine for some period of time? (Assuming I'm giving the machine a push to start the motion initially) Inevitably, provided the above would work, friction would lead to an eventual slowing. With that said is there anyway to postpone that slowing for a reasonable period of time? Can magnetic shielding be used in any fashion to maintain the motion for longer?


Vortices and chemical potential in topological superconductors

I am trying to read up some review articles about Majorana physics in topological material, but I am not really familiar with the condensed matter terminology (with condensed matter in general I should say) since I come from a high-energy background, so I come across quite a lot of vocabulary and visualization issues. From the little that I know from semi-conductor physics, a gap is an energy difference between two bands such that a particle cannot go from one band to the other without being given at least that amount of energy. With that picture of band structure in mind, I don't really know what a gap potential is, nor how it can have vortices in it. For instance, in a $p_x + i p_y$ superconductor in 2D, it is said that Majorana fermions appear in vortices in the superconducting pairing potential, or when the gap is closed by variations in the chemical potential. I am wondering if there is an intuitive picture of what a "pairing potential vortex" is, without getting into solving the BdG equations for the superconductor, and on how Majorana fermions actually appear in them? Moreover, another question that pops into mind is related to the use of chemical potential in the Hamiltonians describing superconductors. Statistical mechanics tells us that the chemical potential is the energy necessary to add one particle to a system from a reservoir, and also conveniently describes the energy costs related to diffusion processes in solutions. How does one interpret the chemical potential in a superconductor then? I have read somewhere that a non-homogeneous chemical potential $\mu(x)$ is a sign of an electric field $E(x) \sim \mu'(x)$, so it seems there would be a relation between $\mu$ and the electrostatic potential, but I don't find anything information that explains the relations between all these quantities in superconductors.


Does a straight water hose issue water at a greater pressure than a Coiled water hose of same diameter and length?

I have a one BHP water pump, the water pressure of a coiled hose connected to the water pump output side was not that great. Would an unwound water hose produce greater water pressure? [Friction Losses?] Thanks, Alan


Is the long range neutron-antineutron interaction repulsive?

I can model this interaction as Zee does in "Quantum field theory in a nutshell". In chapter I.4 section "from particle to force" he uses two delta functions for the source. The integral gives $E=-\frac{1}{4\pi r}e^{-mr}$ I consider $J=\delta^{(3)}(x-x_1)-\delta^{(3)}(x-x_2)$. Im assuming that this represents a particle and an antiparticle. I have done the calculations quantum and classically and I obtain $E=\frac{1}{4\pi r}e^{-mr}$ So as this simple model is used to describe the long range nucleon interactions I think I could conclude that this is the potential interaction between a neutron and an antineutron, a repulsive Yukawa potential. I want to know If Im right. Textbooks dont talk about this and Zee seems to say that the force is always attractive.


Why does the fundamental mode of a recorder disappear when you blow harder?

I have a simple recorder, like this: ![enter image description here](http://i.stack.imgur.com/eYO9G.jpg) When I cover all the holes and blow gently, it blows at about 550 Hz, but when I blow more forcefully, it jumps an octave and blows 1100 Hz. What's the physical difference between blowing gently and blowing forcefully that the recorder suddenly jumps an octave and the fundamental mode is no longer audible?


How Galaxy is formed?

Given the distance among stars (the most massive objective in the space) is so huge, the difference of order of magnitude is about 7. And also, since gravity is such a weak force, how is it likely for gravity to shape galaxy the way it is? Would it make more sense to say galaxy is collapsing? I would like to know if my reasoning is improper, where goes wrong, and how to test my reasoning or your reasoning with experiment?


Electrodynamics textbook that emphasizes applications

Which textbook in Electrodynamics which emphasizes practical applications and real life examples would you recommend for undergraduates?


Determining the Hodge numbers of some orbifold examples

I'm currently reading about complex geometry in order to get a feeling of how to determine the Hodge numbers, e.g. of certain orbifold constructions. Since I'm a physicist with no deeper mathematical knowledge in algebraic geometry and complex geometry, I apologize for the following possibly trivial questions. In the following I'm referring to two examples of the string theory book by Becker², Schwarz. Example 1 (p. 368): $T^4/\mathbb{Z}_2$, i.e. $z_1 \sim z_1+1$ and $z_2 \sim z_2 + i$, where the $\mathbb{Z}_2$ isometry is generated by $\mathcal{I}:(z_1,z_2) \to (-z_1,-z_2)$. First of all, it is easy to see that there is no invariant $(1,2)-$form, hence $h^{1,2}=0$ and that $dz_1 \wedge dz_2$ is the only $(2,0)-$form, i.e. $h^{2,0}=1$. Moreover, there are four invariant $(1,1)-$forms (they're obvious, so I'm not gonna write them down) which contribute to $h^{1,1}$. But there are in total also $16$ fixed points of the orbifold which can be blown up à la Eguchi-Hanson. It is claimed that $h^{1,1}=4+1 \times 16 =20$ because there is one $(1,1)-$cycle for every blown up singularity. Sorry for this questions: How do I see that there is just one $(1,1)-$form for every singularity? And why aren't there cycles from the EH-Space that contribute to $h^{2,2}$ and the other Hodge-numbers? Example 2 (pp. 372): Now we consider $T^2 \times T^2/ \mathbb{Z}_3$ with the action $(z_1,z_2) \to (\omega z_1, \omega^{-1}z_2)$ with $\omega =\exp(2\pi i/3)$. One has then $9$ singularities.   In this case there are $2$ invariant $(1,1)-$forms. It is claimed that there are now $2$ $(1,1)-$cycles for every blow-up, i.e. $h^{1,1}=2+ 2 \times 9 =20$. Why are there now two $(1,1)-$cycles for each singularity? And why aren't there other cycles contributing to the remaining Hodge-numbers? I would be happy if someone could explain me the intuition and the maths behind this counting. psm


The relation between energy quanta of an Einstein solid and the equipartition value of heat capacity

Consider an Einstein solid with quantized energy values $U=q\epsilon$ and $N$ oscillators. I calculated some values of an Einstein solid numerically through a function in R (at the bottom of the post). In an exercise I'm supposed to estimate $\epsilon$. Apparently the answer is that I can estimate it by showing that $kT/\epsilon$ is approximately $1/3$ when the heat capacity reaches half of its maximum value (this value is called the equipartition value). I find this conclusion really confusing, since the energy quanta don't have anything to do with the heat capacity (or do they?). The **exact** wording in the solution of the exercise is > Note that in the high-temperature limit the heat capacity approaches the > result predicted by the equipartition theorem, $C=Nk$ (since there are two > degrees of freedom per oscillator). Below $kT=\epsilon$. however, the heat > capacity falls off dramatically. (...) The value of $\epsilon$ can be > estimated by noting that the heat capacity reaches half of its equipartition > value at $kT\approx\epsilon/3$. Could anyone please explain how does conclusion works? I can't get a grasp on understanding the relation between the energy quanta and the equipartition value of heat capacity. The function I wrote to produce a table uses column names such as `C_Nk` where I in fact mean $C/Nk$, R just doesn't handle slashes in function names so well:               install.packages('gmp');install.packages('Rmpfr');library('gmp');library('Rmpfr')     EinsteinProperties <- function(N){          q_total=100          q=0:q_total          Omega=formatMpfr(chooseMpfr(N+q-1,q),digits=2)          S_k=formatMpfr(log(chooseMpfr(N+q-1,q)),digits=4)          kT_eps=rep(0,length(q))          C_Nk=rep(0,length(q))          for(i in 0:q_total){              if(i>0&&i<q_total){                  kT_eps[i+1]=formatMpfr(2/(log(chooseMpfr(N+(i+1)-1,i+1))-log(chooseMpfr(N+(i-1)-1,i-1))),digits=2)              }else if(i==0||i==q_total){                  kT_eps[i+1]=0              }          }          for(i in (0:q_total)){              if(i==1){                  C_Nk[i+1]=formatMpfr(2/(2/(log(chooseMpfr(N+(i+2)-1,i+2))-log(chooseMpfr(N+(i)-1,i))))/N,digits=2)              } else if(i>1&&i<q_total){                  C_Nk[i+1]=formatMpfr(2/(2/(log(chooseMpfr(N+(i+2)-1,i+2))-log(chooseMpfr(N+(i)-1,i)))-2/(log(chooseMpfr(N+(i)-1,i))-log(chooseMpfr(N+(i-2)-1,i-2))))/N,digits=2)              } else {                 C_Nk[i+1]=0              }          }          table <- cbind(q,Omega,S_k,kT_eps,C_Nk)          layout(matrix(1:2,1,2))          graph_S_k <- plot(q,S_k,type="l",xlab="U/eps",ylab="S/k",main="Entropy",xlim=c(0,100))          graph_C_Nk <- plot(kT_eps[2:q_total],C_Nk[2:q_total],type="l",xlab="kT/eps",ylab="C/Nk",main="Heat capacity",xlim=c(0,2),ylim=c(0,1))          return(table)     }     EinsteinProperties(50)


Energy of damped harmonic oscillator begins to increase with very large Q in numerical integration

I have numerically integrated the (reduced) homogeneous equation of a damped harmonic oscillator in order to see how the error propagates. $$\frac{d^2 X}{d\phi^2} + \frac{1}{Q}\frac{dX}{d\phi}+X(\phi) = 0$$ where $\phi = 2 \pi \tau$ and $\tau$ is the reduced time In the process I have also extracted the logarithm of the energy and plotted it against time, for different values of the quality factor, Q. One can expect the energy to decay slower for larger Q. $$ E = P^2 + X^2$$ where $P = \dot{X}$ What I found is that for very large values of Q, the energy of the oscillator begins to **increase**. I checked my code several times and I couldn't see anything wrong with it so I suspected that it purely something to do with numbers so I began playing around with them, but first here is the graph. ![enter image description here](http://i.stack.imgur.com/9LTVF.png) What I found is odd; say I'm using a step size of $1 \times 10^{-n}$ . If Q's magnitude matches or exceeds $10^{n}$ the energy begins to increase! What is causing this? **What is the connection between Q and the step size?** **Also, should the energy be oscillating?** (If you zoom in on the red and green lines on the graph you will see that it is oscillating much slower there.) * * * Here is how I am integrating the equation for those interested:   1. Let $\dot{X} = P$    2. Thus original equation becomes $\dot{P} + \frac{1}{Q} P + X = 0$    3. Now we have two 1st order DEs.    4. Using discrete steps we can roughly find the next point of $X$ and $P$ by considering that $\dot{X} = \frac{dX}{d\phi} = \frac{X_{n+1} - X_n}{\Delta \phi}$ and approaching the other eqution in the same manner.    5. We can then rearrange for $X_{n+1}$ and $P_{n+1}$ and create a code that calculates those points, given some basic initial conditions.


Does the current acceleration of universe imply that our universe is open? If the universe is closed, from the Friedmann's equation, the acceleration of universe wouldn't be possible, would it be? (Of course, except for the very earlier time of inflation era.)


Why does the water in the toilet move around so much on stormy days?

On calm days, the water in the toilet looks completely still. But when it's rainy and windy out, the water looks like it moves and pulsates. Why is this?


Calculating dragging frame of satellite orbiting Earth

Say there is a satellite polar-orbiting the Earth at 600km. How much would the satellite be dragged additionally due to dragging force? Such that $$ \Omega = \dfrac{r_s \alpha r c}{\rho^2(r^2 + \alpha^2) + r_s \alpha^2 r sin^2 \theta}$$ such that: $r_s = \dfrac{2GM}{c^2}$ is the Schwarzchild radius. $\alpha = \dfrac{J}{Mc}$ $\rho^{2} = r^{2} + \alpha^{2} \cos^{2} \theta$ $\Delta = r^{2} - r_{s} r + \alpha^{2}$ I'm a bit confused how I'm supposed to use this formula as $\Omega$ denotes angular speed not distance doesn't it? Thank you so much for any help


Reconciling "The Big Crunch" with the 2nd Law of Thermodynamics

Assume "The Big Crunch" scenario (the universe will collapse to a singularity). In this case, I think of the entire universe as an isolated system; in the "Big Crunch" scenario, it seems to me gravity itself would act as the "walls" of the system, keeping all the matter and energy inside. Thus, by the second law of thermodynamics, entropy will increase. However, it seems to decrease (everything is being smashed together). I know black holes get around this through the emission of Hawking radiation, but this is the emission of radiation from black holes to the universe. In the big crunch scenario, by definition, there is no universe for the radiation to escape to because the universe itself is collapsing, taking all radiation with it.


Interferometer problem about periodic fringe patterns

Please,kindly help me; my teacher assigned me problem set, and she never went over anything about interferometers in class. Also, there is almost no information about about how to solve interferometer problems in my book. I have no idea how to set this problem up, and I have no intuition concerning how I should solve it... Can someone help me set this problem up and explain it? I know if this example is solved, I will be able to solve the rest of the problems: Sodium D lines, of wavelength(a)=589.0 nanometers and wavelength(b)=589.6 nanometers are used on a Michelson interferometer. When the first mirror moves, these lines periodically appear and disappear. Explain, in detail, this phenomenon and write an equation that would allow you to know how much the mirror must move to make the lines reappear and disappear one time.


Euler-Lagrange equation for continuous systems

I'm having a little trouble with wrapping my head around a part of a method which is fairly 'new' in some fashions to me. I imagine it should be fairly obvious, but I am not seeing something at the moment. Let's take the Lagrangian density to be $$\mathcal{ L}={1\over{2}}\left(\mu\dot{\phi}^2-Y(\partial_x\phi)^2\right),$$ where $\mu$ is mass density of rod and $Y$ is Young's modulus. One "should" be able to put this into the equation $${\partial\over \partial t}\left({\partial \mathcal{L}\over{\partial(\partial_t \phi)}}\right)+{\partial\over \partial x}\left({\partial \mathcal{L}\over{\partial(\partial_x \phi)}}\right)={\partial \mathcal{L}\over{\partial \phi}}$$ to get the equation of motion for the rod $$\mu\ddot\phi=Y\partial_{xx}\phi,$$ right? The trouble is I'm not exactly seeing where to go with the LHS of the E-L equation applied to $\mathcal L$ at the moment. I'm hoping someone would be so kind as to demonstrate that in a few lines.


Any resources to learn how to interpret formulae/diagrams?

Are there any materials exploring intermediate levels (of any sub-subject) which give plain English descriptions along side formulas/equations//diagrams/notations?


Space Warps & Frame Invariance

The Alcubierre Warp Drive in Higher Dimensional Spacetime (H. G. White & E. W. Davis), among other sources, proposes that the x-axis symmetry paradox for Alcubierre-style warp drives can be resolved by interpreting the warp field as producing a scalar boost on a pre-existing velocity. If that is the case, how can it obey frame invariance? In other words, how does the boost "know" which velocity, in which inertial frame, it's supposed to be multiplying?


What force keeps electrostatic charge from exiting a metals surface

I understand that in electrostatics all charges are 'static' due to equal forces acting upon them. This then leads to the fact that the arising electric field must be perpendicular to the metals surface at the surface. What i don't understand is what force is counteracting the electric field. I have read that there is a potential well but am not sure how this works. Could someone point me in the right direction? Thanks.


How does QFT help with defining measurement in Quantum Mechanics?

I learned that a quantum system has an "overall state", a vector in a Hilbert space. That Hilbert space can be decomposed in a basis of "basic states". For example if in the Universe there is only a free electron with its own spin, and I pick an axis $z$, there is an infinite basis of "position+spin", let's say $|(0,0,0);1/2>, |(0,0,0);-1/2>, |(0.1,0,0);1/2>\ldots$ I cannot obviously write all possible positions, but that was to give the idea. Some $\psi (\vec{r},s)$ contains the coefficients for all the elements of this basis and is given. This is the "overall state" of my "one-electron Universe" I was said that there is "measurement" that makes the $\psi$ collapse to one of the "basic states" and can give the observer the result (position + spin in the $z$ axis) Since a measurement involves an interaction with some sort of detector I figured out that I would need QFT (the possibility to have multiple particles and interaction) to understand measurement and add a detector into my Universe. I thought that the dynamic of the interaction between the particles of the detector and the electron could act in a way that the $\psi$ collapsed (I hoped that even in QFT there was something analogous to the original $\psi$ of the electron). Now I couldn't study QFT but from what I grasped it is much more complicated and the reason of the collapsing of the wave function is not explained by an "interaction". The wave function itself is subjective and depends on the knowledge of the observer. So I am lead to believe that QFT does not contain anything that explains the collapse of the wave function. Is it true? I know that I presented a rather subjective view. I hope that they are shared by many and that an answer that points out my misunderstandings could help others. I think that many people in the quest for a more "classical" explanation of QM resort to thinking that when they would see how the interactions work "really" so as to describe a fully fledged measurement equipment (photons, QED, QCD...) they would figure that out.


LIGO sensitivity in terms of minimum received power per area

I've been looking at LIGO figures for gravitational wave sensitivity here and it seem to be displayed in an adimensional strain ratio (which I assume that is more or less equivalent to metric perturbation $h_{\mu \nu}/g_{\mu \nu}$, correct me if I'm wrong) What I'm wondering is how can I convert this sensitivity measure into something like watts/area, as a measure of energy density flux of gravitational waves.


A fundamental equation for solitary wave and dimension analysis

According to the scalar Field theory we write Lagrangian as $$\mathcal{L}=\frac{1}{2}\partial^\mu \phi \partial_\mu \phi -\frac{m^2}{2}\phi^2 -\frac{\lambda}{4!}\phi^4 \tag {1}$$ What I want to do is to get a equation of solitary wave (solition) from the above Lagrangian which will lead for solving many solitary problems like life-time, dimension analysis etc. My confusion is choosing the potential because it may vary and depend on the author's selection. What are the physical constraints on choosing an appropriate potential? Another rather technical expectation is solitary wave is nonlinear equation so how will I write down the equation as 2nd order differential equation with respect to time?


What happens between two harmonics?

I know that standing waves in a simple harmonic system occur when the "echo" of a wave overlaps completely with the original wave at a certain point in time, doubling the amplitude. And then, because both waves have the same speed and frequency but opposite direction, they move away from eachother until they cancel eachother out, and then keep moving until they overlap again to amplify the signal. But what happens halfway between two harmonics? Would it be like a half standing wave, or would the signal cancel out?


Noether's Theorem: Foundations

I'm wondering on what principles Noether's theorem foots. More precisely: _The action is a functional on the fields only. Why do we consider then variations of the space time too? In principle careful considerations, however, seem to untangle them as special field variations. So what's going on here truly?_


Does Archimedes law depend of lack of matter?

I would like to know if Archimedes law take in equation the lack of matter. Fluid density at 1. An object with density of 0 (or near 0) is put in liquid, it's fixed. The external sphere where liquid is inside is fixed too. The lack of matter change the attraction in all the sphere itself, no ? Density change with depth, no ? How forces can be calculate ? ![e](http://i.stack.imgur.com/Jn1KB.jpg)


Speed of sound in non-newtonian fluids

Recently I saw some videos of non-newtonian (shear-thickening) fluids like corn starch mixed with water(sometimes known as oobleck), where the fluid is placed on top of a speaker cone and starts to create odd shapes as it becomes thicker in some places. Now I wondered what would happen in the following situation: Suppose a room with reflective walls is fully filled with such a fluid. On one side of the room a sound is produced by a speaker. On the other side of the room is a microphone. How will a sound wave propagate through such a substance? Does the fluid increase its viscosity before, during or after a wave moves through it? What will the speed-of-sound be in such a case? I've read that the speed of sound is much faster in solids than in liquids, but how is that for this kind of substance?


Is there a way to fill Tank 2 from Tank 1 through Gravity alone?

I am a newbie in water system design but I am currently faced with the exact situation below on my land, and I need to know whether gravity alone is sufficient in order to fill Tank 2 from Tank 1, as I already experienced backflow. Please have a look at this image: ![enter image description here](http://img819.imageshack.us/img819/184/qhzf.png) _Note: let's not worry about how Tank 1 is being filled as it probably does not matter - I just made sure I never used a pipe diameter smaller than 0.5" between Tank 1 and Tank 2._ As you can see:   * My water source is the overflow of Tank 1   * I have put a 1" diameter pipe in the first 'downhill' section of my path   * The second, longest section of the path is a pipe of 0.5" diameter   * The system is 'powered' through gravity alone **Questions**   * Is there any missing variable (eg path length?) on my drawing in order to resolve this system? (What are the important variables?)   * The flow at the source (the overflow of Tank 1) can be nearly 0. How does this variable influence the system? (Can a larger flow at the source help going up the 'uphill' sections between the tanks?)   * Is the 'head' variable important at all?   * I used a bigger pipe for the first section in an attempt to 'make more weight' and create sufficient pressure to go up the small uphill that follows. Should I limit the use of that bigger pipe strictly to the downhill section, and does it actually make sense to have used a bigger pipe at all?


Solving straight-line motion question for time

I apologise in advance if this question doesn't appeal to the advanced questions being asked in this Physics forum, but I'm a great fan of the Stack Exchange software and would trust the answers provided here to be more correct than that of Yahoo! Answers etc. A car is travelling with a constant speed of 80km/h and passes a stationary motorcycle policeman. The policeman sets off in pursuit, accelerating to 80km/h in 10 seconds reaching a constant speed of 100 km/h after a further 5 seconds. At what time will the policeman catch up with the car? The answer in the back of the book is 32.5 seconds. The steps/logic I completed/used to solve the equation were:   \- If you let x equal each other, the displacement will be the same, and the time can be solved algebraically. Therefore:   $$x=vt$$ As the car is moving at 80km/h, we want to convert to m/s. 80/3.6 = 22.22m/s   $$x=22.22t$$ As for the policeman, he reaches 22.22m/s in 10 seconds.   $$\begin{aligned} x &= \frac12 (u+v) t \\\ &= \frac12 \times 22.22 \times 10 \\\ &= 111.11 \mathrm m \end{aligned}$$ The policeman progresses to travel a further 5 seconds and increases his speed to 100km/h.   100km/h -> m/s = 100 / 3.6 = 27.78m/s. $$\begin{aligned} x &= \frac12 (u+v) t \\\ x &= \frac12 \times (22.22 + 27.78) \times 5 \\\ x &= \frac12 \times 50 \times 5 \\\ x &= 250 / 2 \\\ x &= 125 \mathrm m \end{aligned}$$ By adding these two distances together we get 236.1m. So the equation I have is:   $$22.22t = 27.78t - 236.1 $$ Which solves to let t = 42.47s which is really wrong.


Mass converted to energy in a common fire?

In a common wood fire such as a campfire, is matter converted to energy or is it simply an exothermic chemical reaction and all the mass can be accounted for in the ash and soot?


Possible spin states?

Given a system of two particles with spin up and down, I have troubles to understand the possible states of this system. I would have normally thought, that the possible states are the tensor products of the states of each particle with one of the other. This would have given us the states that are listed in Wikipedia under: substituting in the four basis states. So why do we define these new states(three with total spin $1$ and one with total spin $0$) that are given by triplet and singlet representations? So maybe you don't know what I am asking so I repeat my question: What is the meaning of these triplet and singlet states, because I would have thought that the four states listed above are the possible states of a system? The reference to what I am talking about: wikipedia reference


4-acceleration of rotating frame

Consider the 3-dimensional Minkowski space $$ ds^2=dt'^2-dr'^2-r'^2d\phi'^2 $$ Now we transform it into a rotating frame: $$ t'=t,r'=r,\phi'=\phi+\omega t $$ Then the metric becomes $$ ds^2=(1-r^2\omega^2)dt^2-dr^2-2\omega r^2d\phi dt-r^2 d\phi^2 $$ Consider a static observer in the rotating frame with coordinate $(r,\phi)=(R,0)$. Could anyone show me how to compute the 4-acceleration of this observer? I am expecting the answer to be $(0,-\omega^2 R,0)$ based on my knowledge of Newtonian mechanics. But my calculation does not give this. I have calculated the 4-velocity to be $$ u=\left(\frac{1}{\sqrt{1-R^2\omega^2}},0,0\right) $$ Then I use the formula $$ a^\mu=u^\nu\nabla_\nu u^\mu=u^t\nabla_tu^\mu=u^t(\partial_tu^\mu+\Gamma^\mu_{t\alpha}u^\alpha)=u^tu^\alpha\Gamma^\mu_{t\alpha} $$ Then it turns out only $\alpha=t,\mu=r$ is meaningful, and $$ a^r=-\frac{2R\omega^2}{1-R^2\omega^2}R^2(1-2\omega^2R^2) $$ which I think is definitely wrong...


Why does milli- mean 1/1000

I suppose this is also an English question, but I'll ask it here first. Why does the milli- prefix mean 1/1000 when it sounds so much like million? According to the internet, this dates back to the 18-19th centuries. Did it make more sense back then? Are we stubbornly holding on to meanings from borrowed prefix's languages (such as milli meaning 1000 in Latin)? Not that I'm qualified in any way to make this kind of statement, but if I were able to rename our unit prefixes, I would shift everything from milli- to femto- down 3 orders of magnitude and eliminate atto-. The resulting pairs would be:               thousand-    kilo    (placeholder)     million-     mega    milli     billion-     giga    micro     trillion-    terra   nano     quadrillion- peta    pico     quintillion- exa     femto     sextillion-  zetta   zepto     ...      Which to me, sounds better. But I suppose I answered my own question by saying that since they are the units everyone uses, social inertia prevents us from changing. Would this be correct?


Glycerol: refractive index & absorption spectra in 0.2-0.4um range

Could anyone suggest where can I find absorption spectra & refractive index of Glycerol? I am specifically interested in UV range, 200-400nm, everything I was able to find out was for standard conditions only... Is there any software which can get these parameters through simulation?


How would normal matter behave under conditions found in the core of the sun?

Considering how low the power density is at the sun's core, I seem not to be able to expect what would happen to matter in case it was thrown inside the sun's core. For example, let's assume an Earth-like planet is placed at the center of the Sun's core, and with the power density of 276.5 $watts$/$m^3$ which is very low to even raise the temperature of the Earth by any noticeable degree. That was one thought. The other thought was that, the temperature in the core is already 15 million degrees K, so any matter there should get close to this temperature quickly enough. So now I'm very confused, like will the planet stay there intact for thousands or even millions of years until it has accumulated enough energy to melt or vaporize ? Or there are other types of energy absorption the planet would experience and thus have a shorter time staying as one piece in the core ? Just for trying to get a practical and numerical answer to this question, let it be : how long can the Earth as a whole survive in the core of the sun ?


Mathematical proof of non-negative change of entropy $\Delta S\geq0$

I understand that we can prove that for any process that occurs in an isolated and closed system it must hold that $$\Delta S\geq0$$ via Clausius' theorem. My question is, how can I prove this in a mathematical way?


Why force $F$ is $ma$ but not $md$ or $mv$? How can I observe and understand "force" in real life?

As a layman, i can calculate approx "displacement" just by observing the moving object. And accurately by using a simple "scale". Similarly, again, I can calculate angle from origin by using displacement in $x$ and $y$ dimensions. Similarly I can use a stopwatch and scale to understand velocity. But when I read about force, first of all it confuses with the english word "force", we use in real life. To some extent I am sure, it has nothing to do with that. So exactly what is it in the real sense. How can a layman, see force as, just like he can see displacement or angle? Or force is just a quantity defined by physicists to simplify the combinations of $ma$, they might be facing every time. And thus came up with term "force" ( which is similar in spelling to english word in oxford dictionary). And lastly, why not force has just been called as something proportional to mass and it's displacement or velocity. Why something at the level of change of velocity has been used to define it.


Regarding the theory of the origin of water on earth through meteorites, why wouldn't the water evaporate on impact?

Water on earth has been theorized to have come through comets trapped inside crystals. But why wouldn't that water evaporate on impact, and wouldn't the atmosphere at that time allow the vapours to escape Earth? Also, what is the current scientific opinion of the validity of this theory ?


$\tau$ pair production question

There's a question on my homework about the process $e^{-} e^{+} \rightarrow \tau^{+} \tau^{-}$. Specifically, it is claimed that the minimum energy required of the colliding positron and electron beams is slightly less than twice the $\tau$ mass, and I am asked to explain this deviation and compute it. In the context of this course, we have only discussed the relativistic kinematics of such processes (which would predict a minimum energy of twice the $\tau$ mass), so I am not sure what might be responsible. I'm grasping at straws here, but might it have something to do with the Coulomb interaction between the opposite charges? Thanks in advance.


Is the potential in Schrödinger equation an operator?

In the Schrödinger equation in the position representation $$ i\hbar\frac{\partial}{\partial t}\Psi(x,t) ~=~[\frac{-\hbar^2}{2m}\nabla^2+V(x,t)]\Psi(x,t), $$ is the potential $V(x,t)$ an operator acting on $\Psi$ or merely a scalar?


Self-inductance of a toroidal inductor

I am trying to determine the self-inductance of a toroidal coil of mean radius $R$ with $N$ loops of radius $a$ with a current $I$ flowing within them. I have calculated the magnetic field by noting that it must be in the circumferential direction (i.e. in the $\hat{\boldsymbol{\theta}}$ direction) and then using Ampéres Law to get: $$\oint_{\partial \Sigma} \mathbf{B}\cdot \mathrm{d}\boldsymbol{\ell}=\mu_{0}I_{\text{enc}} \implies \mathbf{B}=\frac{\mu_{0}NI}{2\pi r}\hat{\boldsymbol{\theta}}$$ I know that we can get the self-inductance $L$ by using the relation: $$\Phi=LI$$ And we know that: $$\Phi=N\iint_{\Sigma}\mathbf{B}\cdot\mathrm{d}\mathbf{A}$$ Where $\Sigma$ is the area of a single loop. However, I am not sure how evaluate the surface integral, I'm not sure if perhaps I should be using the mean radius $R$ to give us: $$\Phi=\frac{\mu_{0}N^{2}}{2\pi R}\pi a^{2}I=\frac{\mu_{0}a^{2}N^{2}}{2R}I$$ Or if I need to evalute the integral in a more complete way?


Gravity through a portal

Ok, this is definitely a silly question and is partly inspired by the last 20 minutes or so of the film _The Avengers_ , so look to that if you can't picture the situation. In the film a portal is opened up above new york, to somewhere in deep space. I.e. from an area where the gravitational field is effectively uniform and field lines would be parallel, to an area with negligible net gravitational acceleration. My question is, assuming this portal has literally linked the two bits of space, it makes sense that the gravity of the earth would be transmitted through the portal in some way, but in what form would the field take on the other side of the portal? Would it behave like a point source, or similarly to a diffraction slits? Would a mass slightly off to one side from the portal experience a force? Let's ignore all the practical factors like the air rushing out into the vacuum and the question of what happens on the "other side" of the portal (the bit you'd be looking at if you flew up from new york to above the portal) although there's plenty of scope for interesting things happening there as well.


Is fire plasma?

Is Fire a Plasma? If not, what is it then? If yes why, don't we teach kids this basic example? ![enter image description here](http://i.stack.imgur.com/MLPSn.jpg) **UPDATE:** I probably meant a regular commonplace fire of the usual temperature. That should simplify the answer.


A photon travels in space for 10 billion years. What are the odds it will arrive here without interacting with a atom on the way?

Space isn't a perfect vacuum and I wonder how an image of a galaxy can travel billions of years without becoming diffused by photon collisions with space matter?


Simple Quantum Mechanics Question about The Commutator of Translation Operators

Say there is $\hat{J} = \exp[-i \hat{p} l/ \hbar]$ and $\hat{U}= \exp[-i\hat{H}t/ \hbar]$, where $\hat{H}$ is time-independent. Can we say anything about $[\hat{J},\hat{U}]$? Is it zero? How do we show this? For example if $\hat{H} = \hat{p}^2 /2m + m\omega^2 \hat{x}^2/2$.


String theory and the SM spectrum

Long ago, I realized this: (super)string theory can NOT give a well- defined/unique prediction of why the electron (muon, tau) or the neutrino (any flavor) masses have the masses we measure. String theory can NOT give a concrete prediction of any SM particle mass (even resonances like the Higgs and other particles). The question is: Is this lack of predictive ability of the mass of the SM "elementary particles" a hint that string theory in its current status is not complete or wrong? Or perhaps even better: **Even when string theory provides a framework in which you can accomodate every Standard Model field, and hence, a priori you can have the SM spectrum, we have not a string theory explanation of the values of the masses and charges. Should a explicit derivation of particle properties like (rest) masses and (bare) charges (like the electron charge value) be "predicted" and "derived" in a more fundamental version of the theory?**


Incorrect IR temperature reading on stainless steel?

Using a Fluke 62 Mini Infrared Thermometer today on a stainless steel pipe with coolant I noticed that it gives incorrect temperature readings. There is a temperature sensor inside the coolant saying about 90° Celsius. I spray painted a spot on the pipe and the IR thermometer instantly gave more correct readings on the painted spot. So why? Why is it the coolant fluid is about 90° Celsius (both the painted spot and the temperature sensor suggest that), while the unpainted clear stainless steel says about 50° Celcius? I.e. apx 55%


Euler Lagrange equation in different frames

Suppose I have an inertial frame with coordinate $\\{q\\}$. Now I define another reference frame with coordinate $\\{q'(q,\dot q,t)\\}$. I obtain the equation of motion in $\\{q'\\}$ in two different ways:   1. First obtain the equation of motion in $\\{q\\}$ by the Euler Lagrange equation $$\frac{d}{dt}\left(\frac{\partial L}{\partial \dot q}\right)-\frac{\partial L}{\partial q}=0$$ and then rewrite the equation in terms of $\\{q'\\}$.   2. First transform $L(q,t)$ to $L'(q',t)=L(q(q',t),t)$ and then obtain the equation of motion $$\frac{d}{dt}\left(\frac{\partial L'}{\partial \dot q'}\right)-\frac{\partial L'}{\partial q'}=0$$ Are the two answers just the same?


Why does quantum mechanics invalidate one of locality and realism?

I am approaching this from an intuitive perspective and I don't speak the language. However, I have been doing a lot of reading about Bell's Theorum and how invalidates either locality or counterfactual definiteness. From my point of view, locality is the concept that there is no way that two particles can interact with each other "instantaneously" at a distance (i.e. faster than light.). Realism/counterfactual definitiveness is the concept that all particles have some true value for all variables at some instant regardless of whether or not that value was observed. Let's assume for the sake of this question that locality is true, which I believe is the current popular model in QM. Assuming locality, why do Bell's Inequalities show that conterfactual definiveness must be false? **Edit to respond to below post:** This is crazy. Let me see if I am following you. I'll use your framework. I'm not really sure what spin is but I'm seeing it as an observable phenomenon that exists as a function of a specific direction in a given particle at an instant of time, and the spin itself at any angle changes (randomly?) over time. A and B are, we are assuming, perfectly correlated so so the spin at angle θ in A is definitely the same as the spin at angle θ in B. In any given electron (let's just take A) we can define some angle θ as the angle between two 'directions' on that electron that have an exactly 99% correlated spin. There's no way we can measure those two directions on the same particle at the same time, but since A and B are perfectly correlated, we can measure A at 0 degrees and B at θ degrees and experimentally we would see that they are the same spin This is crazy. Let me see if I am following you. A and B are, we are assuming, perfectly correlated so so the spin at angle θ in A is definitely the same as the spin at angle θ in B. In any given electron (let's just take A) we can define some angle θ as the angle between two 'directions' on that electron that have an exactly 99% correlated spin. There's no way we can measure those two directions on the same particle at the same time, but since A and B are perfectly correlated, we can measure A at 0 degrees and B at θ degrees and they will be the same spin 99% of the time. Similarly, if we measure the spin at θ in A and at 2θ in B, we will see that these two values are the same spin 99% of the time. Assumingly, this will work for any two angles that are exactly θ apart -- with enough measurements they will be correlated exactly 99% of the time. However, if we measure the spin at 0 in A and at 2θ in B, they are only correlated 96% of the time. If V0 is at most 1% off from V1, and V1 is at most 1% off from V2, V0 should be anywhere from 0-2% off from V2. But, we find, according to QM and experimentation, it is not! It is less heavily correlated than that. This is where I lose you. Why does saying no to realism resolve this issue?


What's so special about wave solutions of EM?

Maxwell's equations allow for wave solutions via oscillations between electric and magnetic field content. Couldn't we generate electric waves also if that solution didn't exists? Imagine there was no magnetic field, only the electric field. If at one point in space we generated an electric field nicely varying in a sinus pattern of constant frequency, wouldn't this propagate with light speed, just like a plane EM-wave of that frequency, even though there was no magnetic field?


Is temperature quantized?

I'm learning quantum mechanics on my own. I've known that energy is quantized and I've started wondering about temperature. From thermodynamics we have: $$U=\frac{3}{2}NkT $$ (for ideal gas, of course)   Both U and N aren't continous, so i think T shouldn't be, too. Is that formula correct also for quantum mechanics?   I'm really sorry because of my language, I'm still working on it.


Quantum Boltzmann Equation

What is the Quantum Boltzmann equation and what does it describe? I think it describes the propagation of electrons and photons but I am not sure.


Relationship between nuclear spin and nuclear magnetic moment?

We know that nuclear magnetic moment can be expressed in terms of the expected value for nuclear spin as: $$\langle\mu\rangle =[g_lj+(g_s-g_l)\langle s_z\rangle]\frac{\mu_N}{\hbar}$$ (Cf. Krane), where $\vec{j}$ is the _total_ angular momentum, $\vec{l}+\vec{s}$. How does the expected $\langle s_z\rangle$ value relate to the $\vec{j}$-component of spin, $\langle s_j\rangle$? Krane mentions that only that value is needed, given that it remains constant.


Electronic filter

Can you explain, please, **step-by-step** how an electronic filter does work? For example, high pass filter. I know It's a trivial things, but I can't get it completely. Don't bring me formula and etc. Just explanation in three words.


Non-Conservative Behaviour of Static Electric Field

Static electric fields are supposed to be conservative in nature and therefore give $0J$ work if traversed over a loop. However in the following problem I got non zero work by a static electric field. How can I explain this work ? ![enter image description here](https://lh4.googleusercontent.com/-S3oWGGy_30Y/UuLvVIXZnFI/AAAAAAAAAhE/kRutOGK9Q88/s1668-Ut/2014+-+2)   **Note :** This is done in field of line charge, diagram shows a cuboid type structure but that is just for differentiating between traversed path and line charge. I am writing some of the important parts/equations here, but I have written the whole derivation here. $$W_{AB} = \frac{q\lambda}{4\pi\epsilon_0} [ln|\frac{(\sqrt{(L)^2+(r_1)^2}+L)(r_1)}{(\sqrt{(\frac{L}{2})^2+(r_1)^2}+\frac{L}{2})^2}|]$$ $$W_{BC} = \frac{q\lambda}{4\pi\epsilon_0}[ln|\frac{(\sqrt{(\frac{L}{2})^2+(r_1)^2}+(\frac{L}{2}))(r_2)}{(\sqrt{(\frac{L}{2})^2+(r_2)^2}+(\frac{L}{2}))(r_1)}|]$$ $$W_{CD} = \frac{q\lambda}{4\pi\epsilon_0} [ln|\frac{(\sqrt{(\frac{L}{2})^2+(r_2)^2}+\frac{L}{2})^2}{(\sqrt{(L)^2+(r_2)^2}+L)(r_2)}|]$$ $$W_{DA} = \frac{q\lambda}{4\pi\epsilon_0}[ln|\frac{(\sqrt{(L)^2+(r_2)^2}+(L))(r_1)}{(\sqrt{(L)^2+(r_1)^2}+(L))(r_2)}|]$$ $$W=W_{AB}+W_{BC}+W_{CD}+W_{DA}$$ $$W = \frac{q\lambda}{4\pi\epsilon_0} [ln|\frac{(\sqrt{(\frac{L}{2r_2})^2+1}+\frac{L}{2r_2})}{(\sqrt{(\frac{L}{2r_1})^2+1}+\frac{L}{2r_1})}|]$$ If field were to behave conservatively $W=0$ must have been satisfied, but unless $r_1 = r_2$, $W \neq 0$. Thus field is not behaving conservatively. PS : This is not a homework question ! I am trying to explain the behaviour.