title: Magnetic forces and generators

text: i'm really struggling to break this question down into manageable chunks and identify which quantities are relevant. I'm also unsure of some of the wording in the question. > A bar of length $20$$\mathrm{cm}$ and negligible mass can slide over two > conducting rails connected to a DC generator producing an $\text emf$ of > $V_{0} = 6V$, connected so as to produce a current as in the figure. The > resistance of the bar is $R = 0.08$$\mathrm{Ω}$, all other parts have > negligible resistance. The bar is connected through a pully to a body of > mass $1.2$$\mathrm{kg}$ the system is immersed in a uniform magnetic field > orthogonal to the rails, as in the figure, whose magnitude is > $1$$\mathrm{T}$. The system is designed in such a way that, after a while, > the body is pulled upward with a constant limiting speed. Compute: > > a) The current flowing in the circuit and the power provided by the dc > generator when the limiting speed is reached. > > b) The magnitude of the limiting speed. > > c) The value of the resistance of the bar corresponding to which the body > does not move at all. I do not know what 'limiting speed' means, I have searched online and through my text book to no avail. I think if I had a clear picture of what that was, i'd be able to make some progress. At a guess; $EMF = VBL$ $V = \frac{EMF}{BL}$ = $30$ $\mathrm{ms^{-1}}$ Is this the limiting speed? When the bar is stationary (or if there was no magnetic field?), the current in the circuit could be found easily via ohm's law. $I = \frac{V}{R} = \frac{6}{0.08} = 75 \mathrm{A}$ I feel like i'm going round in circles if I say that the bar is moving so there should be an induced current in the bar. I think that would violate conservation of energy, since it would keep getting faster and faster. So I think it's correct to say that a current of 75 Amps makes the bar move at 30 meters per second and likewise, if you were to move the bar at 30 meters per second, you would induce a current of 75 Amps. I don't know if my reasoning makes sense up to this point, but I don't know how to take it much further. I'd really appreciate a helping hand, thanks! Jacob.


title: How does electrostatic force affect electronic devices?

text: How does electrostatic force generated by two seperate plates having opposite charges affect electronic devices? I know that magnetic fields have some harmful effects to electronic devices but I am not sure if electrostatic forces have these harmful effects since magnetic fields are generated because of the movement of electrons, however, electrostatic fields are generated just because of the presence of charges.


title: Quantum information science references

text: I was hoping you guys could recommend reading material on Quantum Information Science. First off, here's my background. Personally, I started reading Ballentine's Quantum Mechanics and I found it be a very consistent book in terms of foundations and absolutely loved it. The treatment is better than most other books on the subject, including the popular ones like Sakurai, Merzbacher. I wondered why there existed this difference. Why were the others rather sloppy? I found my answer when I discovered my new favorite author, Asher Peres. As I read his works, I found this quote in an excellent paper (Rev. Mod. Phys., 76, 2004) "Quantum mechanics is used by theorists in two different ways. It is a tool for computing accurate relationships between physical constants, such as energy levels, cross sections, transition rates, etc. These calculations are technically difficult, but they are not controversial." The other group is tackling the foundations. I was drawn towards this group and as I read more of Peres' works, I got absorbed with the idea of information in QM. I then read many of C. Fuchs' works and followed on with those in the Perimeter Institute, where this sort of research seems to be hot. Coming from the background of Peres and Ballentine, my gut reaction to books which talk of collapse or simultaneous measurements or quantum mechanics of individual systems as opposed to ensembles is to shut them. Only slowly am I overcoming this, because I find that most books are very sloppy and that if I wish to learn any more, I cannot afford to do so. I am trying to be as open as possible. Currently, I am reading the book by Busch, Lahti and Grebowski, notes on Arxiv by Keyl and also the notes by Preskill. If anyone has any suggestions, recommended reads, I'd love to know!


title: About change in velocity of a light wave as it enters a different medium

text: $\dfrac {sin\theta1}{sin\theta2}=\dfrac {v1}{v2}=\dfrac {n2}{n1}$ I understand this equation, but what is the velocity of a light wave going through air and what is the velocity/change in velocity as it enters standard glass of roughly 1.5 refractive index?


title: What does the appearance of a classical particle fundamentally reduce to?

text: I've been reading an article that describes what seems to be a classical particle as a regularity in the global wavefunction over a quantum configuration space: > When you actually see an electron trapped in a little electron trap, what's > really going on, is that the cloud of amplitude distribution that includes > you and your observed universe, can at least roughly factorize into a > subspace that corresponds to that little electron, and a subspace that > corresponds to everything else in the universe. So that the physically real > amplitude distribution is roughly the product of a little blob of amplitude- > factor in the subspace for that electron, and the amplitude-factor for > everything else in the universe. ...which looks sensible enough, but I cannot figure out what is actually meant by this configuration space; specifically, what is the identity of an individual point in that space? A simplified picture - mentioned in the above article - tells us that an individual point corresponds to something like "a photon here, a photon there (etc., for every photon), an electron here, an electron there, etc. for every species of elementary particles". But this can't be quite right: if an electron corresponds to a blob of amplitude in the configuration space, a single point in that space can't correspond to a position of this blob... and besides, what would a dimension of that space even be, since the number of particles in the universe isn't fixed. An "explanation" offered in the article is as follows: > A single point in quantum configuration space, is the product of multiple > point positions per quantum field; multiple point positions in the electron > field, in the photon field, in the quark field, etc. But what does this actually _mean_? As I'm given to understand, in quantum field theory one has a wavefunction(al) defined over all configurations of a specific field, so if one jumps over to the real world where space-time is filled with many fields, is it possible to speak of the quantum configuration space over which the universal wavefunction is defined as a space where every point corresponds to some configuration of every quantum field, and if so, can this picture still be used to see an electron as a roughly independent part/blob of this giant amplitude distribution or not? I've been gnawing at this for two days now and I'd really appreciate some help here...


title: How does gravity behave outside the geoid?

text: I was watching a program about the geoid recently and the relative "height" of the Himalayan plateau gave me an idea. It makes sense for the geoid to be "highest" (is that the correct term?) there as there is more mass underneath you than in other places, however I remembered from physics that your distance from the center of the earth effects gravity as well. So if you were on the peak of Mount Everest what would have a stronger impact on your weight as compared to say sea level at a comparable latitude (to control for the spin of the earth), there being more mass underneath you, or you being further from the center of the earth? In other words, would you feel slightly lighter or slightly heavier? I tried working it out myself, but I haven't done any calculus in over fours years, so I couldn't figure it out.


title: Naive questions on Goldstone modes and a possible duality relation?

text: For example, let's consider a 1D spin-1/2 ferromagnetic (FM) Heisenberg chain $H=-J\sum_{i=1}^{N}\mathbf{S}_i\cdot\mathbf{S}_{i+1}$ with periodic boundary conditions. Now we want to study its low energy excitations via the following two approaches: (1)Jordan-Wigner (JW) transformation, we get $H\approx H_f=\sum \omega_kf_k^\dagger f_k-\frac{NJ}{4}$, where $f_k$ are JW fermionic operators; (2)Holstein and Primakoff (HP) transformation, we get $H\approx H_b=\sum \omega_kb_k^\dagger b_k-\frac{NJ}{4}$, where $b_k$ are HP bosonic operators. Where in both of the above two expressions, $\omega_k=J(1-\cos k)$ (setting lattice constant unity). Then the ground states of $H_f$ and $H_b$ are the vacuum states and both **exactly equal to** the _exact_ FM ground state of the original Hamiltonian $H$. And the ground state energy of $H_f$ and $H_b$ (the constant terms $-\frac{NJ}{4}$) is **exactly equal to** the _exact_ ground state energy of the original Hamiltonian $H$. Note that the approximation $\approx$ indicates that we have assumed the spin fluctuation (i.e., $\left \langle \hat{n}_i\right \rangle\ll 1$) to be small and hence dropped the higher order terms (i.e., interactions between JW fermions or HP bosons), which is justified since the ground states are the vacuum states containing no JW fermions or HP bosons ($\left \langle \hat{n}_i\right \rangle=0$), implying that this approximation is at least self consistant. Furthermore, $\omega_k\approx Jk^2/2$ as $k\rightarrow 0$ corresponds to the Goldstone mode. If we believe that the above two pictures are both correct, then I get some questions: (1) Can the elementary excitations of a system be either fermions or bosons which may depends on the theory we adopt? Is there some _duality relation_ or deep connection between the fermionic and bosonic approaches? (2) From $H_f$, can we infer that the gapless Goldstone mode is a fermion? As we always say a Goldstone boson instead of a Goldstone fermion. Thank you very much.


title: Decoherence without time?

text: Decoherence is a phenomenon that provides a part of the explanation of why quantum systems and classical systems behave differently. What I understood from decoherence so far is that it requires time because it simply takes a (somehow very short) while for a large quantum system to transit to classical behaviours through decoherence. Now, the status of time in quantum gravity is not clear, and it could very well occur that time is an emergent quantity. So I thought decoherence should occur at time scales where time has already emerged. Yet I found this assertion on www.decoherence.de > Decoherence explains also how the Schrödinger equation of general relativity > (the Wheeler-DeWitt equation) may describe the appearance of time in spite > of being time-less" I would like to understand how can decoherence be active in the timeless domain of the Wheeler-deWitt equation to allow the emergence of time, as decoherence seems to require time itself to happen. (Handwaving arguments are OK, but references are welcome!)


title: Is Jupiter a failed star?

text: In my physics lessons, my teachers have always been keen to tell my class that Jupiter is considered a 'failed star' by scientists. Is this true? In my own effort I wondered if maybe this could just be being regurgitated from an outdated physics syllabus that still considers the Solar System to have nine planets. From that thought onward, through my research on the Internet, I haven't found people referring to Jupiter as such and people always call it a planet rather than a brown dwarf. Furthermore, it's my understanding that brown dwarfs possess more mass than Jupiter suggesting to me that Jupiter possesses too little mass for fusion to even be plausible. So am I correct in thinking that Jupiter is 'only' a planet, or are my physics teachers correct in saying it is a failed star (and if so, why)?


title: Continuous Variable Entanglement Measure for the Statistically Mixed State

text: Can anybody tell me, which is the best entanglement measure for the Continuous Variable Entanglement of a Statistically Mixed State ? I have read that Schmidt decomposition is not valid in this situation. Hence I cannot use von- Neumann entropy of the reduced density matrix. The system that I study has dissipation terms and the states are no longer statistically pure states.


title: How does space affect the human body (no space suit, no space craft)

text: How does "outer space" affect the human body? Some movies show it as the body exploding, imploding or even freezing solid. I know space is essentially a vacuum with 0 pressure and the dispersion of energy makes it very cold. So are the predictions above accurate?


title: What happens to light that falls into a black hole?

text: When light enters a black hole, what happens to it? I imagine the photons will either fall into the singularity, or the light will orbit just inside the event horizon indefinitely. (Some background to my question...) I was reading a description of what someone falling into a black hole would experience, which said they wouldn't realize they had just passed the event horizon. I imagined that there would be an awful lot of star-light that's been shining onto the black hole for billions of years, so the doomed astronaut would see a large flash of light as they cross the event horizon. I also imagined that when a black hole loses mass and the event horizon shrinks, there would also a flash of light as the captured light managed to escape. More likely is that I've made an error in my reasoning.


title: Can a DIY balloon become a danger to others?

text: A couple of years ago I remember reading that a DIY group sent a $25 balloon 20km into the air. This made me think for a while because if they lift about 5Kg that high It would generate more or less 1000kJ of kinetic energy upon impact. Checking the orders of magnitude, in Wikipedia, that’s equivalent of "a 2 tone vehicle at 116 km/h". Which I think is a lot. There must be some factor that I'm not thinking of (ex: air resistance, hydrogen to kg lifted ratio, etc...) preventing bad intentioned people from dropping that weight whenever they want with a low frequency radio transceiver and wrecking havoc with this apparently simple method, what are those factors?


title: Why are binary objects so critical to astronomy?

text: There are a lot of astronomical objects out there to study but binary objects seem to get more attention. Why is this? What makes binary objects (stars, Kuiper Belt Objects, black holes, galaxies, etc.) more important or special than lone objects?


title: Proof: Conservation of a 4-vector in one frame implies conservation in another

text: I came across a proof that says if a 4-vector $P$ is conserved in one inertial frame $$P_{before}=P_{after} \text{ (sum)}$$ then Lorentz transforming to another frame gives $$P'_{before}=\Lambda P_{before} = \Lambda P_{after} = P'_{after}$$ However, I am worried that the terms in the sums $\Lambda P_{before}$ and $\Lambda P_{after}$ will not be all at the same instant of time. But in order for $\Lambda$ P to be deemed conserved the terms have to be simultaneous. How does one resolve this problem? Thanks.


title: Timelike Loop Spaces as Projective Null Twistor Spaces

text: Let $\mathcal{M}$ be a spacetime, and let $\Omega\mathcal{M}$ denote the loop space of the spacetime. My idea is that the set of all closed timelike curves of $\mathcal{M}$ forms the projective null twistor space. The following is an excerpt from my own essay: > Consider the projective null twistor space of $\mathcal{M}$. This space has > light rays equivalent to points, and light cones as Riemann spheres. Let us > define $\Omega_t\mathcal{M}$ as the timelike loop space of $\mathcal{M}$, > i.e., the set of all timelike curves in $\mathcal{M}$. Consider a closed > timelike curve in $\mathcal{M}$. Restrict the curve so that one can deform > it into a line. Define a subclass of spacetime $\mathcal{M}$ as the class > $\mathbb{A}\subset\Omega_t\mathcal{M}$ of closed timelike curves in > $\mathcal{M}$ such that all curves > $\varsigma\in\mathbb{A}\subset\Omega_t\mathcal{M}$ are homeomorphic to a > portion of the real line $\mathbb{P}$ which lies on the null cone of any > point. This class is trivial, as homeomorphisms are equivalence relations. > Thus, the class of CTCs forms a light ray in spacetime. This corresponds to > a point in $\Omega_t\mathcal{M}$. Similarly, consider a region on the > spacelike hypersurface such that the region is bounded by the sections of > light rays that fall on the hypersurface. This corresponds to a Riemann > sphere in $\Omega_t\mathcal{M}$. Thus, $\Omega_t\mathcal{M}$ is isomorphic > to the projective null twistor space of $\mathcal{M}$. > > Evolve $\Omega_t\mathcal{M}=\Sigma_T$ (This relation holds since Riemann > spheres on $\Omega_t\mathcal{M}\mbox{ and }\Sigma_T$ correspond to sections > of light cones on $\mathcal{M}$ and light rays $\mathcal{M}$ to points on > $\Omega_t\mathcal{M}\mbox{ and }\Sigma_T$. Therefore, by the above theorem > and the transitivity of homeomorphisms, $\Omega_t\mathcal{M}=\Sigma_T$.) > with respect to time, where $\Sigma_0\times\mathbb{R}^1=\mathcal{M}$, with > respect to time to get the the ``positive'' part of $\mathcal{M}$, i.e., the > part $\mathcal{M}_+$ of $\mathcal{M}$ with all points $x(t)$ in > $\mathcal{M}_+$ having $t>T$. Does this method of thinking make sense? I have done some research on the topic and it seems to be consistent with earlier established results.


title: caustics on droplets on glasses formed by streetlights

text: Suppose you're out at night, and it's rainy, and your glasses are covered by water droplets, and you chance to look at a streetlight.   * Are the ridged caustics of light seen at the edges of these droplets described by the Airy functions?    * Is there a good way of simulating their appearance?


title: Would a solution to the Navier-Stokes Millennium Problem have any practical consequences?

text: I know the problem is especially of interest to mathematicians, but I was wondering if a solution to the problem would have any practical consequences. Upon request: this is the official problem description of the aforementioned problem.


title: Eigenfunctions corresponding to degenerate spectra

text: It is well knwon that an eigenstate can be obtained by superposing wavepackets. In other words, if $\Psi({\bf x},t)$ is the solution of the time dependent Schroedinger equation for an initial wavepacket $\Psi({\bf x},0)$, the eigenstate $\psi_n({\bf x})$ corresponding to the eigenenergy $E_n$ is related to $\Psi({\bf x},t)$ by \begin{equation} \int\Psi({\bf x},t)e^{iE_nt/\hbar} dt=2\pi \hbar\sum_m a_m\psi_m({\bf x})\delta(E_m- E_n)\propto\psi_n({\bf x}), \end{equation} where the coefficients $a_m$ are defined by $\psi({\bf x},0)=\sum_m a_m\psi_m$. Now comes my question, what happens when the spectrum is degenerate (or quasidegenerate)? i.e. when two eigenvalues $E_n$ and $E_{n+1}$ are equal (or very close to each other). Does the same relationship hold?


title: Uniqueness and existence of polygonal orbits through a spherical shell

text: Say we have a spherical wire mesh raised to a negative voltage. Then let's say we release a proton from near the surface, and away from the surface, at some angle and speed. Also, imagine that the proton will always miss the mesh wires and pass through the surface. The field in the center of this sphere is zero, so the proton travels in straight lines there, and outside of the sphere the proton travels in elliptical orbits with one focus located at the center of the sphere. If the proton exits 3 times and winds up back _at the same point it started_ , I imagine its trajectory would look something like this, where the dotted lines are the other part of the elliptical orbit that did not travel. ![Triangular orbit](http://i.stack.imgur.com/5dom9.png) I wonder about a few things about this:   * Is the above idea possible in the first place?   * would it be possible to have a 2-exit orbit in the same sense?   * Is there only _one_ combination of angle and velocity you can release it at to get a N-gon orbit?   * Alternatively, is there a relationship between angle and speed that fits such an N-gon orbit? The above picture is imagining a "rounded" triangle orbit. If the last case is true (that there isn't a single unique N-gon orbit), then I wonder if there are varying degrees of roundness of the corners you could produce. If not, I imagine the orbital geometry would force all N-gon orbits to be similar to each other. I can't answer this easily to myself, and it seems such an interesting question.


title: How to calculate the temperature of a receiving point from concentrated solar power?

text: I'm exploring a DIY project about solar power and have found very few resources that can explain, in a not too simple but not too advanced way, how to go about calculating important values in solar power, specifically concentrated solar power. So maybe someone here could help me get started or show me where I can find answers to this question. I am wondering how I can calculate the temperature that a receiving point of focus can attain from concentrated sunlight. The following example shows the basic variable factors that I am aware of. Suppose:   * There is a material in the shape of a dish with a surface area $x$ and a reflectivity factor $r$   * There is a clear cloudless sky with sun directly above the dish and the current air temperature is $y$ (e.g. $25^\circ\mathrm{C}$ or $77^\circ\mathrm{F}$)   * The dish reflects light onto a receiver which is at a perfect focal point   * The receiver has heat absorption efficiency of $a$ and insulation efficiency of $i$. Is there a simple formula that can put these factors (and probably others I am unaware of) together to calculate or approximate the temperature $q$ the receiver can reach after a time $t$. For example, something like $$q = rxyt + ai \tag{no doubt this is wrong}$$ Ignore for simplicity (if you like) some of the details for achieving a geometrically optimal shape, and the different amounts of heat depending on the location of the dish on the earth and the time of the day. I am most interested in the qualities needed by the reflecting material. **Besides geometric efficiency, is reflectivity the main factor in the performance of the reflecting material?** I have fairly beginner/amateur level knowledge of physics. In this area I have read chapters about optics in a first year level college physics book. But optics seems like such a vast area, at the moment I am just interested in finding out the details of just a small area of it.


title: If Fine Structure Constant in not uniform in space what would that imply for cosmic background?

text: Recently J.Webb submitted paper which has extraordinary claim - that Fine Structure Constant is different in different directions in space! He (with others) measured $\alpha$ using quasar spectrum absorption method in different directions and found small variation. There is huge debate about this finding, but lets suppose it is true. Do you think spacial variation of Fine Structure Constant could have some influence to the Cosmic microwave background radiation? Because $\alpha$ influences the Electromagnetic interactions which is responsable for atom formation. And neutral atoms were the key for transparent universe in its early development sages. Could this be observed?


title: Probability and lifetime

text: What is the relation between the lifetime of a particle and the probability of decaying that particle? Here it says that the probability of survival is exponential if the decay process is a Poisson process: $P(t) = e^{-\frac{t}{\tau}}$ However, this function is not normalized so it cannot be a probability distribution. If the decay lifetime of a particle is $\tau$, what is the probability of detection a decay an time t?


title: If electromagnetic induction generates a potential difference in a loop, where are the "high" and "low" potentials?

text: In AP Physics, I learned that when a changing magnetic flux generates an electromotive force around a loop, and emf is measured in volts, where are the "high" and "low" points to measure the potential difference? Also, how is electromagnetic induction consistent to the conservation of energy?


title: Quick question on perturbation theory

text: Suppose we have a particle in an infinite potential well, with $V(x) = 0,\space 0< x < a $ and infinity everywhere else. Now suppose we have a perturbation on the LHS of the well: $V_1(x) = v, \space 0 < x < \frac{a}{2}$. By first order perturbation theory, all energy levels are shifted by the same amount: $$\Delta E = \langle E_n|v|E_n\rangle = \frac{v}{2} $$ Ok all is good. By considering the perturbed wavefunction: $$ \psi_n = \phi_n + \sum_{n \neq k} \frac{\langle \phi_k|V_1|\phi_ n\rangle}{E_n - E_k} $$ How do I make use of the condition that $v << E_2 - E_1 $ to show the results above is correct?


title: Is the universe expanding at a speed of almost $2c$?

text: I've been told nothing can travel faster than the speed of light. Therefore, from my vantage point the diameter of the universe is increasing at a rate of $2c$. Are there any flaws in my thinking?


The Berry Connection is $$A_\mu(R)=-i \langle \Psi(R) |\partial_\mu \Psi(R) \rangle$$ which allows us to parallel transport a state indexed by $R$. We can integrate the Berry Connection to get the Berry Phase, and we can differentiate the Berry Connection to get the Berry Curvature. Can the Berry Connection be derived from a metric? As a prototypical example, I'm thinking about how the Christoffel Symbols in General Relativity (GR) can be derived from the metric tensor. Also, I think for every connection, there exists a metric for which the connection is a Levi-Civita connection. However, is there a natural and physical metric that induces the Berry Connection? In this presentation, the great and powerful Haldane relates "quantum distance" to the Berry Curvature, but it doesn't look like you can derive the Berry Curvature from the quantum distance in the same way curvature and the metric are related in GR.


title: Calculation of the differential of the entropy

text: In this review (for those who wants a precise reference see page 8 eq 21), the Author says that: \begin{equation*} S=-\sum_{i}P\left(i\right)\ln P\left(i\right) \end{equation*} and using the Maxwell-Boltzmann distribution \begin{equation*} P\left(i\right)=P_{MB}\left(i\right)+dP\left(i\right) \end{equation*} \begin{equation*} dS=-\sum_{i}dP\left(i\right)\ln P_{MB}\left(i\right)=-\sum_{i}dP\left(i\right)\ln e^{-\beta E\left(i\right)}=\beta\sum_{i}dP\left(i\right)E\left(i\right) \end{equation*} However I tried to do the detailed calculation but here is what I get: \begin{equation*} dS=-\sum_{i}dP\left(i\right)\ln\left[P\left(i\right)\right]-\sum_{i}P\left(i\right)\frac{1}{P\left(i\right)}dP\left(i\right)=-\sum_{i}dP\left(i\right)\ln P\left(i\right)-\sum_{i}dP\left(i\right)= \end{equation*} \begin{equation*} =-\sum_{i}dP\left(i\right)\ln\left[P_{MB}\left(i\right)+dP\left(i\right)\right]-\sum_{i}P\left(i\right)+\sum_{i}P_{MB}\left(i\right) \end{equation*} Can someone help me to get the right result? Thanks!


title: Flux cutting and flux linking?

text: I am a bit confused about the difference between flux cutting and flux linking when talking about magnetic fields and induced EMF. I was originally under the impression that flux cutting was when there was relative motion between a conductor and a magnet and linking was when there was a change in the magnetic flux density. After reading, it seems that flux linking is when a magnet is moving and a conductor is still whilst flux cutting is the other away round. This makes no sense to me as depending on the reference frame either could be happening so they seem to be indicating the same thing. If this is however the case what do we call it when the magnetic flux density is changing and there is no relative motion?? (a definition of both would be helpful) * * * Extra: On reading the answers given I think I understand, but just to make sure, would these staments be correct:   1. When a magnet falls through a long metal tube, the field lines of the magnet cut the tube. This is flux cutting.   2. When the flux of the primary coil in a transformer changes, the flux linked with the secondary coil changes. This is flux linking.    3. A metal conductor moves through the magntic field of a magnet and cuts its field lines. Flux cutting.   4. An AC-generator coil spins in a magnetic field, changing the magnetic flux through the coil. Flux cutting or flux linking?? (A bit unrelated but in the expression: $$Flux=BA$$ Is $A$ the cross sectional area of the solenoid?)


title: Using Optic Lens to Simulate Distance

text: I have an actual scenario, where I have to align a particular electronic optical device to two objects in the distance (2-3 km away), forming a right angle at the point where the optical device is. The problem is I have a limited space and would like to do this in a smaller area, maybe in a pitch of the size of a soccer field. I would like to know if it is possible to affix a set of lens in front of the optic device and trick optic device to "see" these two points in the pitch as if they are further away than they actually are. What could be a possible solution to this?


title: Mirror problem of radiation pressure

text: If two perfect mirrors are placed facing one another and they are in proximity, and photons (don't ask me how) are traveling between them and toward one of them, what is to keep the radiation pressure from reaching incredible amounts? I might be way off of base here because I am new to this field. I heard that the radiation pressure doubles on a mirror. If this is the case, a laser beam focused on a mirror at any angle will cause the mirror to accelerate one of two ways-- no more. This also violates energy conservation and thus isn't valid.


title: Law of conservation of matter

text: If scientist have made small particles of matter then why do we still haw the law of conservation of matter? Is it because the few particles don't make a noticeable difference in our life?


title: About higher spin gauge theory

text: I'm going to listen to a talk about Vasiliev's higher spin gauge theory. Before that I want to know more about the background.This is an introduction. Could anybody give a more detailed introduction of this kind of theory? What is the corresponding conserved charge to higher spin particles?


title: Sound travel problem

text: Here's an example of my question to make my explanation a bit easier. Say a decent loudspeaker plays a tune at loud volume 100m away from me and another speaker plays the same tune at lot lower volume 1m away from me. Say the tunes will be just as loud in my ears, and if I'm understanding this correctly we could measure the decibel and record the sounds and they would be equal as well (if we ignore the fact that certain wavelengths travel better through air than others). But a person standing a further 100m away would only hear the loudest speaker (now 200m away), and not the low volume one (101m away). The dB and the sounds wouldn't be identical anymore.To me this doesn't make sense as sound is just air, and I would believe equally loud sounds at one place would fade equally with the same distance added. So my question, why am I wrong (why isn't it like this in reality)? Please forgive me for my explanatory problems as English is not my native language.


title: Index of Refraction in Metal: Approximating Complex Perturbation

text: If you consider waves in a metal, you can write the index of refraction for the metal as, $$ n^2 = 1 - \frac{\omega_p^2}{\omega^2} $$ I am interested in what will happen if the index is perturbed by some small complex quantity, $$ n^2 = 1 - \frac{\omega_p^2}{\omega^2} - i\epsilon $$ In the low frequency limit, this usually corresponds to an attenuation. To compute quantities that depend on $n$ (such as absorption), I would like to expand $n$ since $\epsilon$ is small. The function is, $$ n = \sqrt{1 - \frac{\omega_p^2}{\omega^2}-i\epsilon} $$ So I write, $$ n = \sqrt{1 - \frac{\omega_p^2}{\omega^2}}\sqrt{1 - i \frac{\epsilon}{1 - \omega_p^2/\omega^2}} $$ For this problem $\omega_p > \omega$, so I rewrite, $$ n = i \sqrt{\frac{\omega_p^2}{\omega^2}-1}\sqrt{1 + i \frac{\epsilon}{\omega_p^2/\omega^2 - 1}} $$ And, since $\epsilon << 1$, I tried to expand this: $$ \approx i \sqrt{\frac{\omega_p^2}{\omega^2}-1}\left(1 + i \frac{\epsilon}{2(\omega_p^2/\omega^2-1)} \right) = \sqrt{\frac{\omega_p^2}{\omega^2}-1}\left(i - \frac{\epsilon}{2(\omega_p^2/\omega^2-1)} \right)$$ As a sanity check, I tried plugging in some numbers. $\omega_p=2$, $\omega=1$, and $\epsilon=0.01$. The expansion is off by a factor of $-1$. I believe it is due to the branch cut of the complex square root. If I place $i$ back into the square root, $$ \sqrt{-1 - i \frac{\epsilon}{\omega_p^2/\omega^2-1}} $$ is in the third quadrant of the complex plane. This means that the square root should be in the fourth quadrant, where I have a positive real part and negative imaginary part (in contrast to my expansion above, where the opposite is the case). I spent a long time hunting for this negative, so my question is, is there a cleaner way to expand the function where I won't have this branch cut issue?


title: Infinite and Finite Square Wells

text: For the **infinite square well** in the first region, **outside the well** : $$\frac{-\hbar^2}{2m}\frac{d^2 \psi}{dx^2} + V(x) \psi (x) = E \psi (x),$$ where you set **$V = 0$**. Rearranging gives $$\frac{d^2 \psi}{dx^2} = -\frac{2mE}{\hbar^2} \psi.$$ We identify $$k = \sqrt{\frac{2mE}{\hbar^2}}.$$ For the **finite square well** we have the same situation (for bound solutions) but we set: $$\alpha = \sqrt{\frac{-2mE}{\hbar^2}}.$$ **Why do we absorb the minus sign into the square root now and not before?** How does this tie in with bound/unbound solutions and parity?


title: Why does a car bonnet (hood) rise when you connect the clutch with a brake on?

text: Is the rotational force to overcome the brakes moved to the opposite effect of moving the car chassis, until the brake is released?


title: Where / what is the source of the force that causes refraction?

text: So we know that as light moves from, say, a glass block to air, it speeds up and refracts, but if there is to be acceleration then surely there must be a resultant force to cause it? So where does that come from? Secondly as the light enters the glass block it slows down, but where does the excess energy go?


title: Electric field strength between 2 positive charges

text: I'm having a lot of trouble trying to solve this problem: Three identical point charges of magnitude $+1.0$ $mC$ are located at the corners of a square that is $1.0$ $m$ on a side. A point charge of magnitude $-1.5$ $mC$ is placed at the remaining corner. What is the electric field strength at a point halfway between 2 adjacent positive charges? My thinking was that you can just find the strength of one charge to the center and just double it because they're identical charges. Why is this method wrong and how do I solve this problem? The correct answer is $8.63\times 10^6$ $N/C$, but I have no idea how to get this.


title: What are the practical consequences of "approximate" quantum cloning with a stimulated emission cloning machine?

text: Say I build some stimulated emission-based cloning machine for photons (e.g. some active laser medium). Alternatively I could do cloning via interference, e.g. a linearly polarized photon could be made to interfere with and copy its state on a circularly polarized photon. Here, the no cloning theorem... as far as I am aware... says that we can never exactly copy the exact quantum state of a photon. Quoting from the abstract of: Lamas-Linares, A., et. al. Experimental quantum cloning of single photons. Science 296, pp. 712 - 714 (2002): http://web.physics.ucsb.edu/~quopt/sci_clo.pdf "Although perfect copying of unknown quantum systems is forbidden by the laws of quantum mechanics, approximate cloning is possible. A natural way of realizing quantum cloning of photons is by stimulated emission. In this context, the fundamental quantum limit to the quality of the clones is imposed by the unavoidable presence of spontaneous emission. In our experiment, a single input photon stimulates the emission of additional photons from a source on the basis of parametric down-conversion. This leads to the production of quantum clones with near-optimal fidelity. We also demonstrate universality of the copying procedure by showing that the same fidelity is achieved for arbitrary input states." Now my question is, what are the practical consequences of "approximate" cloning with optimal fidelity (assuming optimal rather than experimental near- optimum fidelity). Does this mean that if we "clone" a polarized photon via stimulated emission, there should be some probability distribution for the cloned photon's frequency or polarization angle? If so, what shape or properties do these distributions have? * * * I think my confusion here is the following (I apologize for the length of this, but I want to illustrate where my ability to reason about the "no cloning" theorem is failing): My understanding is that the "no cloning" theorem says, for example, that we cannot simply inject a single photon with unknown polarization and/or frequency into an optical cavity with an active gain medium, and then expect to be able to copy its information content perfectly, even if we somehow stop the experiment after a single stimulated emission event and block all spontaneous emission events. Why? Because we would need _a priori_ information to properly set up the optical cavity or gain medium in order to perform the copying procedure. It wouldn't work for just _any_ polarization angle or frequency. However, doesn't stimulated emission exactly copy both the polarization and frequency information of the stimulating photon? And, within reason, given the eigenstate distribution of the medium a photon is being extracted from via stimulated emission, shouldn't this process work for a wide range of photon frequencies and arbitrary polarization angles? Even if the efficiency of stimulated emission is very low, what stops us from e.g. running the experiment until a stimulated emission event occurs (as opposed to a spontaneous emission event), and then isolating and individually measuring each photon? Is it the inability to be able to tell for sure that a stimulated emission event has occurred, as opposed to a spontaneous emission event, that leads to the satisfication of the "no cloning" theorem without requiring some degree of error for the stimulated emission copying process? If so, does the lowest achievable probability of spontaneous emission set a hard bound on how well we can simultaneously measure both the polarization and frequency (among other properties) of a photon's state vector? Also, does this tell us that there is a fixed lower-bound error rate for determining a photon's properties regardless of how many properties we measure (since we can individually measure multiple clones produced by the same photon)?


title: Radio-dating and the age of the earth

text: I understand there are a few radio-dating methods to determine the age of the earth, uranium-lead to name one (maybe not the best though). The ratio Pb206 \+ Pb207 to U allows you to find when the uranium came into existence. What I don't understand is how this relates to the age of the earth. Can't the uranium have been rambling for 500 million years in space before lumping together with other elements to form the earth? Which would make the age of the earth in this case 4 billion years instead of 4.55 billion. Or is it the same as with C14 dating: the C14 is formed in the higher atmosphere through interaction of cosmic rays with nitrogen atoms. This stops when the carbon enters living material, so we know then the decay starts. Is the radioactive uranium created by cosmic rays until it hits earth? But the early earth didn't have an atmosphere which would stop the cosmic rays, did it? So I'm a bit confused. How do we know the uranium formed at the same time as the earth?


title: Band gaps: are they at the centre or at the edge of the Brillouin zone?

text: Reading about electronic band structures, I came across the following: > Band gaps open at the edges of the Brillouin zone (BZ), since that is where > the Bragg scattering occurs. I am slightly confused, as most band structures show a gap opening at $\Gamma$ which is at the centre of the BZ. What am I missing?


title: Electrical analogy for stress and strain

text: It feels like the relation between stress and strain (and other mechanical properties) is analogous to that of some electrical properties (voltage and current?). I'm comfortable with electrical engineering and trying to read up on mechanics and viscoelasticity. Could someone help my understanding by drawing some analogies or pointing me to such a description?


title: Magnetic field by a solenoid?

text: I am really not able to understand why is it that the current,while finding the magnetic field in the solenoid (ideal one), is not $I$ the current flowing throught the solenoid but $nI$, where $n$ is the number of turns per unit length. p.s:Using **Amperian loop**.


title: The viscous force between the layers of liquid is same, then why there is variation in the velocities of its layers?

text: I have learned in my textbook that when the liquid flows the bottom layer of the liquid never moves because of friction, but the upper layers move with increasing velocities how it is possible if the viscous force between all these layers is same


title: Physical Significance of Fourier Transform and Uncertainty Relationships

text: What is the physical significance of a fourier transform? I am interested in knowing exactly how it works when crossing over from momentum space to co ordinate space and also how we arrive at the uncertainity relations and how do we interpret them physically. P.S.: Links to articles, papers or book recommendations are welcomed as long as it is relevant to the topic.


title: How can we detect cosmic background radiation?

text: From what I understand, CMB is the left over radiation from the Big Bang. As all matter, including the Earth, was made during the Big Bang and then as the universe expanded that matter/energy got further and further apart, but as that matter can't move faster than the speed of light then we should be behind the CMB which travels at the speed of light. So from my understanding the CMB would have passed us and most of the matter of the universe a long time ago so we shouldn't be able to detect it. Also on a side note how is the CMB constant; shouldn't it be a flash? Sorry if the question doesn't make much sense. I am a student so please use layman's terms.


title: Proof for $p=\gamma_Pmu$

text: As I'm reading about Relativistic Momentum, my book states the following: $$p=m \frac{\Delta x}{\Delta t}=m\frac{\Delta x}{\sqrt{(1-u^2/c^2)}\Delta t}=\frac{mu}{\sqrt{1-u^2/c^2}}=\gamma_Pmu$$ "Whether this is the 'correct' expression for p depends on whether the total momentum P is conserved when the velocities of a system of particles are transformed with the Lorentz velocity transformation equations. The proof is rather long and tedious..." I'm interested in seeing the proof that they are describing as "long and tedious."


title: Simple question about the tidal force (Leibniz's notation confuses me)

text: _I started going through Taylor and Wheeler's **Spacetime physics** (standard textbook on special relativity). This is from exercise 2.8._ Basically we're dropping a bearing ball from a **315 m** height above Earth. Simultaneously, we're dropping another one initially exactly **20 m** above the first one. We neglect air friction. **g** changes with respect to height, and the lower ball (let's call it **L** ) accelerates more than the upper one ( **U** ). And so it is quite intuitive that the distance between them will slightly increase. Now the goal is to show that the increase **Δy = 2 mm** when L reaches the ground. _The point of the exercise is to see how neglectable this tidal force is in the free-float reference frame of the two balls.._ The text explicitly says the fall takes **8 seconds** , which is in line with the fact that **y=gt²/2** (since **y=315 m** and **g≈9.8 m/s²** ). But of course, **g** is not constant, so we have to start with: **g = F/m = (GM/rₒ²)rₒ²/r² = gₒrₒ²/r²** It is asked to take "the differential **Δg** ". Now I am from a pure mathematical background, so I don't really use Leibniz's notation or any of these hand-wavy differential thing. I write **g'(r)**. But here I suppose this is about writing **g'(r) = dg/dr** (and ignore what is a function and what is a number). And then do: **dg/dr = -2gₒrₒ²/r³** and: **dg = -2gₒrₒ²dr/r³** So I suppose they want: Δg = -2gₒrₒ²Δr/r³ And then get the solution Δy (it should be 2 mm) with **Δy = ½Δgt²**. I find this confusing. Are they expecting me to plug Δg in there and just substitude with numerical values? What do I do with r and Δr? And what about t? Use the **8 s** given in the text? Isn't all of this an approximation anyway? **gₒ = 9.8 m/s²** **rₒ = 6.37 ⋅ 10⁶ m**


title: Do semiclassical GR and charge quantisation imply magnetic monopoles?

text: Assuming charge quantisation and semiclassical gravity, would the absence of magnetically charged black holes lead to a violation of locality, or some other inconsistency? If so, how? (I am not asking about how magnetic monopoles explain charge quantisation.) I am looking to fill in the blanks in the claims found in this answer to "Is it possible to separate the poles of a magnet?" according to which "magnetic monopoles certainly exist [assuming] charge is quantized. This follows only from the semiclassical behavior of black hole decay, and so does not require unknown physics ... you can't forbid them, at least not for macroscopically sized black holes, without ruining the theory [of General Relativity]." Related claims are found in this answer to "Why do physicists believe that there exist magnetic monopoles?" according to which, if a black hole cannot carry magnetic charge, it would result in a "violation of the laws of locality". (This refers to "any consistent theory of quantum gravity", which may or may not require more assumptions than the other claim.)


title: Freezing point of water with respect to pressure.

text: I know when the pressure is reduced, the boiling point of water is reduced as well. But how does the pressure affect the freezing point of water ? In a low pressure environment is the freezing point more then 0 C or less ?


title: Solutions of PDEs in different coordinate systems

text: Suppose we have a PDE, for example the Helmholtz paraxial equation: $$ \nabla_\perp^2A+2ik\frac{\partial A}{\partial z}=0 $$ Solutions depend on the coordinate system we are using, i.e. we obtain Hermite-Gaussian modes if we use Cartesian coordinates or we obtain Laguerre- Gaussian modes if we study the equation in cylindrical coordinates. I know there is a relation between Hermite-Gaussian modes and Laguerre-Gaussian modes. In general my question is: if we study an equation in different coordinate system, can we always find a relation between the solutions? Can I be sure there is a relation between the solutions?


title: Writing an arbitrary operator in bra-ket notation

text: An annoying fact about my physics textbook (Griffiths' _Introduction to Quantum Mechanics_ ) is that it introduces bra-ket notation without telling us how to _use_ it. So I have a two-part question for SE:   1. In general, how do I write an operator written in functional form (say, the Hamiltonian, $-\frac{\hbar^2}{2m}\nabla^2+V$) as something in the form $\sum|a\rangle\langle b|$?   2. In particular, one of the questions my professor gave me told me (without explanation) that the Hamiltonian of the finite double well could be written $|L\rangle \langle R| + |R\rangle \langle L|$ for the purposes of calculating tunneling probabilities. How can I reproduce this result? Details for #2: The text of the question reads: A box containing a particle is divided into a left and a right compartment by a thin partition. If the particle is known to be on the right side with certainty its state is represented by the state vector $|R\rangle$. This is basically a particle in a box but where we have chosen to neglect spatial variations in either half of the box. ... The particle can tunnel through the partition; this tunneling effect is characterized by the Hamiltonian $H=\Delta (|R\rangle\langle L| + |L\rangle\langle R|)$ where $\Delta$ is a real number with dimensions of energy. Essentially, I don't understand where that Hamiltonian comes from. I can see that it is something tunnel-y (in the sense that it's doing _something_ with mappings from one state from the other) but not how to rigorously generate it.


title: How to superimpose Wood-Saxon and Coulomb potential?

text: I have just written a simple simulation that models the tunnel-effect of alpha-particles for $^{212}$Po and $^{238}$Ur. In this simulation, I approximate the potential of the nucleus by a simple square well. Now I'm thinking about improving the simple model to a more complex one. My idea is to superimpose the **Wood-Saxon** potential with the Coulomb-potential. The result should look something like this picture: ![enter image description here](http://i.stack.imgur.com/YPB1B.png) My question is: how do I actually get there? Do I just add the Woods-Saxon and the Coulomb-potential? **FOLLOW UP** @dmckee, thanks, it worked and only needed little tweaking. you were absolutely right! ![enter image description here](http://i.stack.imgur.com/I2Pa9.png)


title: Why is second moment of area often referred as moment of inertia?

text: In many references, mostly civil engineering, the second moment of area is referred as Moment of Inertia. Is that "really" correct? From my understanding, moment of inertia is analogous to mass in rotational motion, not the second moment of area.


title: What is the relationship between Elasticity and Compliance?

text: Compliance is like elasticity of hollow tube. Elasticity is less for instance for arteries so they are less compliant. There seems to some sort of relationship between compliance and elasticity. Based on Tyler's comments:   * Compliance is the derivative of strain with respect to stress (or derivative of displacement with respect to force).    * Elasticity is like opposite of stiffness. about which I am not sure. One figure about compliance for vein and artery: ![enter image description here](http://i.stack.imgur.com/jwuWL.png) where I think the slope of the curve (gradient) is the compliance of each curve. **What is an approximate relationship between the elasticity and compliance?**


title: Why coercivity increase in after depositing organic on top of Ferromagnet

text: I had deposited Fullerene on top of Cobalt thin film which result in increase of coercivity.


title: Runge Kutta Method for a Lindblad Equation

text: I am solving a Lindblad equation for a dissipative Harmonic Oscillator. My Hamiltonian is time dependent, My Lindblad Equation can be written as \begin{equation} \frac{d\rho}{dt}=\frac{[H(t),\rho]}{i\hbar}+D(\rho,a) \end{equation} where the last part $D(\rho,a)$ represents the Linbald operator due to the dissipation. Considering Hamiltonian is constant over the interval $t_{1}$ to $t_{1}+dt$. I used the Runge-Kutta fourth order (RK4) method in each time interval and integrated the above equation. I would like to know whether my procedure could be justified? I have seen methods which refer to the interaction picture then apply the RK4 method. I think the method that I mentioned is even more simple, but I would like to know its validity. My mathematical justification is the following \begin{eqnarray} U(t)=\exp{\left(-\frac{i}{\hbar}\int_{0}^{t}H(\tau)d\tau\right)}\\\ U(t,t+dt)=\exp{\left(-\frac{i}{\hbar}\int_{t}^{t+dt}H(\tau)d\tau\right)}\\\ \end{eqnarray} Now I suppose, in the given tiny interval my Hamiltonian is constant, hence I can move it outside the integral and I get, \begin{eqnarray} U(t,t+dt)=\exp{\left(-\frac{i}{\hbar}H(t)\int_{t}^{t+dt}d\tau\right)}\\\ U(t,t+dt)=\exp{\left(-\frac{i}{\hbar}H(t)dt\right)}\\\ \end{eqnarray} Now I can use the Taylor expansion and use the Runge-Kutta to integrate the function. But every time the $U(t,t+dt)$ operator changes. Is my method legitimate?


title: Electrostatics kinematics equations (Coulombs Law)

text: If there are two similarly charged particles $Q$ and $q$ of masses $M$ and $m$, a Coulomb force is applied on each one (action-reaction). Suppose that $Q$ is stationary and we are shooting $q$ with an initial velocity $\vec{v_0}$ from $\infty$ into the field. I have, as well, figured out that by rearranging Coulombs Law and the Fundamental Law of Mechanics we end up with the following differential equation: $$x''(t) [x(t)]^2 = \frac{kQq}{m}$$ because: coulombs law states that $$F = \frac{kQq}{[x(t)]^2}$$ and 2nd Newton's Law: $$F = ma = m x''(t)$$ By rearranging we get: $$m x''(t) = \frac{kQq}{[x(t)]^2} \Leftrightarrow x''(t) [x(t)]^2 = \frac{kQq}{m}$$ We also know that its velocity is zero when $x = \frac{2kQq}{m{u_0}^2}$ (from Work-Energy Thm) Is the general solution of this differential equation (2nd order non-linear ODE) the solution of the displacement over time function? If yes, what is it!?


title: What causes acceleration of particles in the expansion section of a De Laval nozzle?

text: A De Laval nozzle has a compression section, where the propellant is compressed (and thereby accelerated) as it moves towards a narrow section (the throat). After the throat, the nozzle widens out again. Here's a reference image. ![](http://i.stack.imgur.com/6Gv6t.png) According to what I've been reading, the propellant is moving at subsonic speeds before the throat, and is accelerated to supersonic speeds as it passes the throat. Furthermore, the propellant _continues to accelerate_ as the nozzle expands again. What I don't understand is: what force continues to accelerate the propellant after it passes through the throat? I've been trying to imagine a single molecule of propellant and thinking about the forces from the particles around it, but can't come up with a force "pushing" from upstream (because it's moving at supersonic speeds), or "pulling" from downstream (because at the minimum there would be 0 opposing force in a vacuum, but certainly not a negative attractive force from downstream). Any ideas?


title: Conformal symmetry of Navier-Stokes?

text: This question is in reference to the paper arXiv:0810.1545   * Can someone help understand this scaling argument and the proof(?) that there is a conformal symmetry in Navier-Stoke's equation? (..am I even reading the result right?..)    * Is one saying that the equations 3.4 encode the full symmetry of equation 1.1? If yes, how? What calculation do I need to do to check that? I am unable to understand how to read 3.4   * Also what role does equation 2.9 play? Are the new symmetries specific to the form 2.9 or they apply to the general 1.1?   * Also 2.9 and 1.1 don't seem to be the "same" as apparently claimed- 2.9 has the time derivative of a0 and also a viscosity-Ricci term. What's going on?


title: Einstein gravity versus Newton's gravity

text: What's the basic difference between the gravity as seen by Einstein, and that by Newton?


title: where to find sample fluid stirring data?

text: I want to find some force data for fluid stirring. Originally I intended to record the data using my force sensor, but it doesn't have the resolution to read stirring force for Newtonian fluid. Unless there's a Newtonian fluid with high enough viscosity to produce a reaction force greater than $0.5\,N$, my sensor can't read it. Is there a place to download some sample graphs for the force of fluid stirring motion?


title: Is there a good explanation for the observation of Martian canals?

text: Martian "canals" have been observed by independent observers after their first description. Now, they are attributed to "optical illusion", but I think that this is not a good choice of word, because an optical illusion should be visible today as well. It would have to be a psychological effect, but it is rather astonishing to have people draw whole maps of non-existent canal systems. Is there a good explanation (optical or psychological) for the observation of Martian canals?


title: "Dear radioactive ladies and gentlemen" - Letter by Wolfgang Pauli

text: In 1930, **Wolfgang Pauli** wrote a letter to Lise Meitner for a convention in Tübingen, considering the problem of beta decay. Does anybody know, where to find the original letter online ?


title: Is sea water more conductive than pure water because "electrical current is transported by the ions in solution"?

text: Apparently, electrical charge is transported by the ions dissolved in water, is this true?


title: What is the probability that a star of a given spectral type will have planets?

text: There is a lot of new data from the various extrasolar planet projects including NASA's Kepler mission on extra-solar planets. Based on our current data what is the probability that a star of each of the main spectral types (O, B, A, etc) will have a planetary system?


title: What is Dark Energy, and how was it discovered?

text: I've heard a lot about Dark Energy, and how it's supposed to be one of the most powerful forces in the universe. What is it, and how was it discovered?


title: What is a proportionality constant? (Planck's constant)

text: I understand that Planck's constant is essentially the ratio between the energy of a photon and its frequency. There are 2 things that im trying to verify:   1. isn't the number that Planck's constant represents just an arbitrary result of the measurement system we use? As in, Planck's constant represents the ratio between 1 Joule and 1 Hertz? So if i assume correctly, Planck's constant is the increase in energy (in joules) that you would get from an increase in frequency of 1hz? Thus if i wanted to I could make an alternative unit for energy which had a 1 to 1 correspondence with hz, and no proportionality constant would be required?   2. So regardless of units used to measure, a fixed percentage increase in one quantity results in an equal percentage increase of the other, correct?


title: The Dual Cloud Chamber Paradox

text: **2012-04-07 Addendum: The Dual Cloud Chamber Paradox** Two 10m diameter spheres $A$ and $B$ of very cold, thin gas have average atomic separations of 1nm. Their atoms are neutral, but ionize easily and re- radiate if a relativistic ion passes nearby. Both clouds are seeded with small numbers of positive and negative ions. The clouds collide at a relative speed of $0.994987437c$, or $\gamma=10$. Both are sparse enough to minimize collisions and deceleration. Both show passage of each other's ions. (a) What do distant observers moving parallel to $A$ and $B$ observe during the collision? (b) Are their recordings of this event causally consistent? The answer to (a) requires only straightforward special relativity, applied from two perspectives. For the traveler moving in parallel to cloud $A$, cloud $B$ should appear as a Lorentz contracted oblate spheroid with one tenth the thickness of cloud $A$, passing through cloud $A$ from right to left. If you sprinkled cloud $A$ with tiny, Einstein-synchronized broadcasting clocks, the $A$-parallel observer would observe a time-stamped and essentially tape- recording like passage of the $B$ spheroid through cloud $A$. The $B$-parallel observer sees the same sort of scenario, except with cloud $A$ compressed and passing left-to-right through cloud $B$. If $B$ is sprinkled with its own set of Einstein synchronized broadcasting clocks, the $B$p-parallel observer will record a time-stamped passage of the compressed $A$ through $B$. So, am I the only one finds an _a priori_ assumption causal self-consistency between these two views difficult to accept? That is, while it may very well be that a recording of a flattened $B$ passing through cloud $A$ in $A$-sequential time order can always be made causally self-consistent with a recording of a flattened $A$ passing passing through through cloud $B$ in $B$-sequential order, this to me is a case where a mathematically precise proof of the information relationships between the two views would be seem like a good idea, if only to verify how it works. Both views after all record the _same_ event, in the sense that every time a clock in $A$ or $B$ ticks and broadcasts its results, that result becomes part of history and can no longer be reversed or modified. It's tempting to wonder whether the Lampa-Terrell-Penrose Effect might be relevant. However, everything I've seen about L-T-P (e.g. see the video link I just gave) describes it as an optical effect in which the spheres are Lorentz contracted at the physical level. Since my question deals with fine-grained, contact-like interactions of two relativistic spheres, rather than optical effects at a distance, I can't easily see how L-T-P would apply. Even if it did, I don't see what it would mean. So, my real question is (b): Does an information-level proof exist (not just the Lorentz transforms; that part is straightforward) that the $A$-parallel and $B$-parallel recordings of dual cloud chamber collision events will always be causally consistent? * * * **2012-03-03: This was my original version of the question, using muonium clocks** My SR question is how to predict length contraction and time dilation outcomes for two interacting beams of neutral particles. The two beams are:   1. In different inertial (unaccelerated) reference frames for the duration of measurement part of the experiment.   2. Intermixed at a near-atomic level so that frame-to-frame photon exchange times are negligible. An example would be two intersecting beams of relatively dense, internally cold muonium atoms.   3. Clock-like even at the atomic level (e.g., decaying atoms of muonium).   4. Part of a single causal unit. By this I mean that there is a sufficient short-range exchange of photons between the two beams to ensure that state changes within each beam have an entropic effect, however small, on nearby components of the other beam. This makes their interactions irreversible in large-scale time. An example would be decay of an anti-muon in one frame transferring energy to nearby muonium atoms in the other frame. Simply illuminating the intersection region with light frequencies that would interact with muonium in both of the beams would be another option.   5. Observed by someone who is herself isolated from the rest of the external universe. Muons generated by cosmic rays and traveling through earth's atmosphere provide a helpful approximation the above experiment. The muons provide the first beam, and the atmosphere forms the second one, which in the case of muons is shared by the observer. Such muons have tremendously extended lifespans, which is explained by invoking time dilation (but not length contraction) for the muon frame as viewed from the atmosphere frame. Conversely, length contraction (but not time dilation) is invoked to describe the view of the atmosphere from the perspective of the muon frame. Since this results in the atmosphere appears greatly compressed in the direction of travel, the muons simply travel a shorter distance, thereby ensuring the same decay result (causal symmetry) for both views of the interacting frames. My question then is this: _For the thought experiment of intersecting, causally linked beams of muonium, what parameters must the observer take into account in order to predict accurately which of the two intersecting muonium beams will exhibit time dilation, and which one will exhibit length contraction?_ **2012-03-04 Addendum by Terry Bollinger (moved here from comment section):** Sometimes asking a question carefully is a good way to clarify one's thinking on it. So, I would now like to add a hypothesis provoked by own question. I'll call it the local observer hypothesis: _Both beams will be time dilated based only on their velocities relative to the observer Alice; the beam velocities relative to each other are irrelevant._ Only this seems consistent with known physics. However, it also implies one can create a controllable time dilation ratio between two beams. I was trying to avoid that. So my second question: Are physical time dilation ratios ever acceptable in SR? **2012-03-06 Addendum by Terry Bollinger:** Some further analysis of my own thought problem: A $\phi$ _set_ is a local collection of clock-like particles (e.g. muons or muonium) that share a closely similar average velocity, and which have the ability to intermix and exchange data with other $\phi$ sets at near-atomic levels, without undergoing significant acceleration. A _causal unit_ $\chi = \\{\phi_0 ... \phi_n\\}$ is a local collection of $(n+1)$ such $\phi$ sets. By definition $\phi_0$ contains a primary observer Alice, labeled $\aleph_0$, where $\aleph_0 \subset \phi_0$. Each $\phi_i$ has an associated XYZ velocity vector $\boldsymbol{v_i} = (\phi_0 \rightarrow \phi_i$) that is defined by the direction and rate of divergence of an initially (nominally) co-located pair of $\phi_0$ and $\phi_i$ particles, with the $\phi_0$ particle interpreted as the origin. The vector has an associated magnitude (scalar speed) of $s_i=|\boldsymbol{v_i}|$. Theorem: If $\phi_i$ includes a subset of particles $\aleph_i$ capable of observing Alice in $\phi_0$, then $\aleph_i$ will observe $\phi_0$ as length contracted along the axis defined by $\boldsymbol{v_i}$. Conversely, Alice ($\aleph_0$) will observe each $\phi_i$ as time dilated (slowed) based only on its scalar speed $s_i$, without regard to the vector direction. This dependence of time dilation on the scalar $s_i$ means that if $\phi_a$ has velocity $\boldsymbol{v_a}$ and $\phi_b$ has the opposite velocity $\boldsymbol{-v_a}$, both will have the _same_ absolute time dilation within the causal unit (think for example of particle accelerators). Analysis: This $\chi = \\{\phi_i\\}$ framework appears to be at least superficially consistent with data from sources such as particle accelerators, where time dilation is observable in the variable lifetimes of particles at different velocities, and where absolute symmetry in all XYZ directions is most certainly observed. Applying observer-relative time dilation to fast particles (e.g. in a simple cloud chamber) is in fact such a commonly accepted practice that I will assume for now that it must be valid. (It is worth noting that while particle length contraction is typically also assumed to apply to fast-moving particles, serious application of length contraction in a causal unit would cause the particles to see longer travel paths that would shorten their lifespans. This is the same reason why muons are assumed to see a shorter path through a length-contracted atmosphere in order to reach the ground.) My updated questions are now: (1) Is my $\chi = \\{\phi_i\\}$ framework for applying SR to experiments valid? If not, why not? (2) If $\chi = \\{\phi_i\\}$ is valid, _what property makes the observer $\aleph_0$ in $\phi_0$ unique from all other $\aleph_i$?_


title: Projectiles Launched at an Angle with unspecified Initial Velocity

text: I'm attempting to do my Physics homework, and I did the first one right, but that problem gave me initial velocity. This problem gives me only the angle relative to horizontal and the distance it covers. Can anyone help me figure out where to start? I've tried but I can't find any formula that I can find initial velocity without having time, or vice versa. Any help would be much appreciated. Here's the problem in full: > A golfer hits a golf ball at an angle of 25.0° to the ground. If the golf > ball covers a horizontal distance of 301.5 meters, what is the ball's > maximum height? (Hint: At the top of its flight, the ball's vertical > velocity component will be zero.) I realize that the vertical velocity component has something to do with it, but I can't figure out where that would fit in.


title: Angular acceleration and linear acceleration

text: I have a small confusion. I learned very recently that all particles of a rotating body have the same angular acceleration but different linear acceleration (same for velocity as well). But how is this possible? I find difficulty understanding this though I know that it is almost simple. Can you please help me understanding this?


title: Practical usage of Moment of inertia

text: A small confusion in understanding the practical application of moment of interia . Why is most of the mass of the wheel concentrated on the rim ? I know that it is to increase the moment of inertia , but what is the practical use of doing so ? Can anyone please help me with this ? Thank you


title: How can a planet gravitationally capture objects?

text: I would expect that any asteroid or other object originating far away but passing near a planet would pick up speed and energy as it approaches, but unless it comes into contact with the atmosphere (or something else orbiting the planet), the object should take a hyperbolic orbit, lose that energy again and ultimatedly escape with the same speed (relative to the planet) as it entered. (Of course, the object's speed relative to other frames of reference could have changed, due to the orbital speed of the planet - eg: the "gravitational slingshot" effect). So do "gravitational captures" imply some contact with the atmosphere or other orbiting objects, to lose enough energy to change a hyperbolic orbit to an elliptical one? (My other hypothesis would be that if there's a large enough moon, an object entering the system in just the right way would become a three body problem and all sorts of strange things can happen.)


title: The triangle inequality in CHSH derivation, where is the triangle?

text: http://en.wikipedia.org/wiki/CHSH_inequality#Bell.27s_1971_derivation ![enter image description here](http://i.stack.imgur.com/RnBHQ.png) ... |X + Y| <= |X| + |Y| ??? CHSH inequality is combined from four relative angles between four polarization axis which do not have any defined length. So where is the triangle, what does X and Y from the triangle inequality correspond to in this CHSH setup? ![enter image description here](http://i.stack.imgur.com/hwINz.png) Instead of one triangle with two sides of certain length, we have four different some things which are called "expectation value". Where is anything here which actually has some length? What does X and Y from the triangle inequality correspond to? How many meters in length is expectation value E(0, 22.5), for example?


title: Formular for interior product, example

text: In Nakahara's Geometry,Topology and Physics, the interior product is defined like this : $$i_X: \Omega^{r}(M) \rightarrow \Omega^{r-1}(M).$$ Where $ X \in X(M)$ and $\omega \in \Omega^{r}(M)$ $$i_X \omega = \frac{1}{r!} \sum\limits_{s=1}^r X^{\mu_{s}} \omega_{\mu_{1}\ldots\mu_{s}\ldots\mu_{r}}(-1)^{s-1}dx^{\mu_{1}} \wedge \cdots \wedge\widehat{ dx^{\mu_{s}}} \wedge \cdots \wedge dx^{\mu_{r}}.$$ I can't make sense out of this. May someone please give me a concrete example? How does $$ i_{e_{x}}(dx \wedge dy) = dy$$ look explicitly?


title: Why does the Hubble parameter keep getting smaller if the expansion of the universe accelerates?

text: Suppose a region of space at a distance D from Earth is escaping from us with the velocity v. Since it seems like the expansion of the universe is accelerating, things at D from Earth should be receding faster and faster. It would seem from the v = HD equation, that H is then getting bigger in time. However, in other sources we can read, that the Hubble "constant" it's getting smaller with time. How come?


title: How does temperature affect a semiconductor band gap?

text: From this link, I've read that > An increased interatomic spacing decreases the potential seen by the > electrons in the material, which in turn reduces the size of the energy > bandgap. Can this statement be explained more clearly?


title: Why is force described as rate of change of momentum?

text: > momentum = mass * velocity Differentiating both sides leads to > force = mass * acceleration since the mass doesn't participate in the differentiation as it is constant. * * * **Is this a sound definition** , or a pointless use of differentiation? Or are there systems (like rockets) where the mass is also a variable wrt time?


title: Showing $K_\pm$ are raising/lowering operators

text: In this post, I have the following operators defined: $$K_1=\frac 14(p^2-q^2)$$ $$K_2=\frac 14 (pq+qp)$$ $$J_3 = \frac 14 (p^2+q^2)$$ I am given $ J_3|m\rangle = m|m\rangle$ and asked to show that $K_\pm \equiv K_1 \pm i K_2$ are ladder operators. My approach (raising operator): $$K_+|m\rangle=K_1|m\rangle+iK_2|m\rangle$$ $$=K_1|m\rangle+[J_3,K1]|m\rangle$$ $$=K_1|m\rangle+ (J_3K_1-K_1J_3)|m\rangle$$ $$=K_1|m\rangle+J_3K_1|m\rangle-K_1m|m\rangle$$ First off, I'm unsure if this is the correct approach, and then I'm also lost on what to do next.


title: What gauge is used in the Lagrangian for a non-relativistic point particle in an electromagnetic potential

text: For the Lagrangian $$L = \frac{1}{2}mu^2 - q(\phi - \frac{\vec{A}}{c}\cdot \vec{u})$$ of a non-relativistic point particle in an electromagnetic potential, what gauge is used for the electromagnetic potential $\vec{A}$ and $\phi$?


title: Capacitors: why is the energy not stored in a magnetic field?

text: When a capacitor is charging, the rate of change $dE/dt$ of the electric field between the plates is non-zero, and from the Maxwell-Ampère equation this causes a circulating magnetic field. Now, since a magnetic field exists, why is the energy of a capacitor only stored in the electric field? Usually the formula for the energy stored goes as $ W = \pi d A \times \frac{1}{2}\epsilon_0 E^2$, where the first term is the volume and latter is the electric field energy density. In Poynting's theorem, the electro- _magnetic_ field energy density is $ \frac{1}{2}\epsilon_0 E^2 + \frac{1}{2\mu_0} B^2 $, i.e. there is also the magnetic field _B_ present. In a capacitor _B_ is non-zero, so why do we not include it in the calculation of the energy stored? In other words, why is the energy stored in a capacitor just $[ (volume) \times \frac{1}{2}\epsilon_0 E^2 ] $ and not $[ (volume) \times (\frac{1}{2}\epsilon_0 E^2 + \frac{1}{2\mu_0} B^2) ] $ ?


title: Why do electric generators need a prime mover?

text: Commonly electric generators need a mechanical torque provided by a prime mover [gas turbine or steam turbine or diesel engine,etc] which is needed for the rotation of the generator rotor. I would like to know why:- 1\. The torque cannot be given directly to the generator rotor provided it can be designed to handle the working medium. 2\. An air gap is provided between the rotor and stator --> why not vacuum? 3\. An electromagnetic torque cannot be given directly to the rotor by a stream of ionised particles so as to eliminate the prime mover requirement.


title: Triple slit experiment

text: If the experiment is done with single photons and a "detector" at one of the slits the interference pattern breaks down. What happens if three slits are used with single photons and a "detector" at, say, the right hand slit. Does the interference pattern occur in the two remaining slits or does the interference pattern breaks down for all the slits ? Thanks for any response


title: Super-String and Anti-Particle

text: I'm afraid this is quite basic question . What I learned from the String theory is that the theory unify the Path-integral, Superposition Principle, and commutation relation .   In that context , I wondered whether the Anti-Particle can be described by the similar way the Superposition Principle is expressed by the string theory . How can the anti-particle be described by the String Theory ?


title: Deriving $F = ma$ - Newton's Second Law of Motion

text: **Context:** In my textbook it is given: 'momentum' short for 'linear momentum': > Mass = $m$, momentum is $p=mv$. In time $\Delta t$, momentum changes by > $\Delta p$, the rate of change of momentum is: > > $$\frac{\Delta p}{\Delta t} = \frac{\Delta(mv)}{t} = m \frac{\Delta > v}{\Delta t}$$ **My Doubts:**   1. Isn't a $\Delta$ sign missing beside the $t$ in the second fraction, and thus it should be $\frac{\Delta(mv)}{\Delta t}$    2. How did they derive the third fraction from the second. I tried a lot but can't seem to get that. **My Work:** I have looked at this question - How does $F = \frac{ \Delta (mv)}{ \Delta t}$ equal $( m \frac { \Delta v}{ \Delta t} ) + ( v \frac { \Delta m}{ \Delta t} )$?, but it's a totally different equation. **My Final Question:** Can someone please clear my doubts about this equation and help me understand how does: $$\frac{\Delta(mv)}{t} = m \frac{\Delta v}{\Delta t}$$ Thanks a lot !


title: Physical consequences of non-trivial quantum states homology

text: The set of quantum states of a finite dimensional system is a complex projective space, whose homology groups are non-trivial http://en.wikipedia.org/wiki/Complex_projective_space#Homology. Has this ever found any application in explaining/predicting some phenomena, or exploited in any way (maybe in quantum computing or quantum information)?


title: Why is there a wave on the water surface when I put something in the water?

text: We've started doing waves in our physics class, but we're doing things very quickly and the teacher doesn't explain anything. And I don't understand why waves work. I was thinking that when I fill a bowl with water, and wait until it's still, and then I put a finger in it, then my finger needs some space where the water is, so the water has to move away, like Archimedes said. So some of that water goes up because it's the only place where there's no finger or the rest of the water, and it's easier for the water to push away the air than the finger or the rest of the water. I think it's only the displacement thing that's working here because I tried putting the finger in the water faster, and the effect was pretty much the same except I spilled a lot of water trying different things. But I don't get it why after the water has formed this something like a wall around my finger (not that I can see this moment; it's too quick), the wall starts to move towards the rim. Why doesn't the water just flow down the wall, like dry sand flows down a sand wall? Then I think the excess water should spread itself over the surface, but it doesn't do that. So how does this work?


title: Confusion about Lie derivative on metric

text: According to this site, the Lie derivative of a $(0,2)$-tensor is $$ \mathcal{L}_XT_{ab}=\partial_XT_{ab}+T_{cb}\partial_aX^c+T_{ac}\partial_bX^c $$ However, according the same website, the Lie derivative of the metric is $$ \mathcal{L}_Xg_{ab}=\nabla_Xg_{ab}+g_{cb}\nabla_aX^c+g_{ac}\nabla_bX^c $$ and if the connection is metric compatible, the first term vanish. My question is, why do we change $\partial$ into $\nabla$ here? In fact, in a note, I see both two expressions for the Lie derivative of the metric. Are they really equivalent? * * * Update: I tried to expand the second equation, and turns out it indeed is the first one. In fact, $$ \begin{align} &\nabla_Xg_{ab}+g_{cb}\nabla_aX^c+g_{ac}\nabla_bX^c\\\ =&X^c(\partial_cg_{ab}-\Gamma^{d}_{ac}g_{db}-\Gamma^d_{cb}g_{ad})+g_{cb}\partial_aX^c+g_{ac}\partial_bX^c+g_{cb}\Gamma^{c}_{ad}X^d+g_{ac}\Gamma^c_{ad}X^d\\\ =&\partial_XT_{ab}+T_{cb}\partial_aX^c+T_{ac}\partial_bX^c \end{align} $$ All the $\Gamma$'s cancel out. But now my question is, is it generally true that when expanding $\mathcal{L}_XT$ for some tensor $T$, we can use either partial derivative or covarient derivative?


title: Artificial gravity on rotating spaceship?

text: One of the possible ways to simulate gravity in outer space is to have a rotating spaceship, so that the centrifugal force experienced provides a gravity-like force. My question is: shouldn't this only work when our feet are touching the floor of the spaceship? Only in that case the floor is providing a contact force to balance the centrifugal force. If we jumped, there is no gravity within the spacecraft so what is it that would make us come back down? Also: imagine we had a shower: what would make the water fall down?


title: Bohr's model of an atom doesn't seem to have overcome the drawback of Rutherford's model

text: We, as high school students have been taught that-because Bohr's model of an atom assigns specific orbits for electrons-that it is better than Rutherford's model. But what Rutherford failed to explain was why electrons don't emit EM waves & fall into the nucleus. I don't see how the introduction of 'atomic orbitals' overcame this defect. Can't it still radiate EM waves?


title: Free particle Schrödinger Equation

text: Some sources give the free-particle solution to Schrödinger equation as $$\psi(x,t) =Ae^{i(kx-\omega t)} + Be^{-i(kx+\omega t)}$$ while some sources give it as $$\psi(x,t) =Ae^{i(kx-\omega t)}$$ Are they the same? I can't see that they are the same thing, especially when we try to find the probability distribution function for both of them. They just don't seem to give us the same probability when we integrate the PDF. What am I missing out here?


title: String breaking in QCD

text: I have trouble understanding string breaking in QCD. I have read an article on arxiv (http://arxiv.org/abs/hep-lat/0505012), and I still don't understand what truly happens. My understanding of string breaking (of a meson - let's say bottomonium) is as follows: We have a quark $b$ and an anti-quark $\bar b$ on distance of $R$ at temperature $T=0$ (this is probably to limit the effects of quark-gluon plasma?). As we "pull" them apart the potential energy increases. At some crytical distance $R_c$ it is more favourable that a pair of quark-antiquark is created from vacuum and thus the final state are two mesons. For calculating the static potential between a quark and antiquark we use correlation function for creating a meson at $t=0$ and destroying it at $t=\tau$. We then use Wick rotation $t \rightarrow -it$ which then transforms oscilating functions into exponentially decaying ones: $$C(R,t)=a_0e^{-\frac{E_0}{\hbar}t}+a_1e^{-\frac{E_1}{\hbar}t}+\ldots \hspace{2 mm}.$$ We interpolate ground state with $e^{-\frac{E_0}{\hbar}t}$ which is just the static potential $V(R)$, since we neglect kinetic energy. My first question is: Does the potential static potential $V(R)$ at temperature $T=0$ saturate or not. Many articles say it follows a Cornell potential, which is $$V(r)=-\frac{e}{r}+\sigma r\hspace{1mm},$$ which grows linearly with distance (and doesn't saturate and thus doesn't show string breaking). My second question is about the first excited state, which should be a final state of two mesons (in this case probably $B$ and $\bar B$ meson). Can we get the energy of this excited state with simple correlation function which I mentioned (energy $E_1$), or must we use the correlation function (which is a matrix) in the article (defined on the first page). They also calculate mixing of states which I don't understand fully. Where do they get the energy of eigenstates? The eigenstates they calculate are:$$|1\rangle=\cos(\theta)|b\bar b\rangle + \sin(\theta)|B\bar B\rangle \\\ |2\rangle=-\sin(\theta)|b\bar b\rangle + \cos(\theta)|B\bar B\rangle\hspace{1 mm}.$$ PS The picture of string breaking is showed on page 19 in the article. I would be extremely thankfull if someone showed me an article which explains (qualitatively and quantitatively) string breaking in simpler language.


title: What is electricity?

text: What is electricity? I know very little physics. I know that there are neutrons, positively charged protons and negatively charged electrons in a particle. I know that electrons have some role in electricity. I have read that flow of electrons is electricity. But I don't understand how flowing electrons can make a bulb light? or a fan rotate? This may be a silly question here, but I couldn't find a better place to ask it.


title: work equals heat in isothermal process only for ideal gas?

text: Is it only for an ideal gas that the work equals negative the heat in an isothermal process or is this general for all kinds of systems? Or put differently, how do I know that for an ideal gas the internal energy only depends on the temperature?


title: How much choice did Einstein have in choosing his GR equations?

text: General relativity was summarised by Wheeler as "Spacetime tells matter how to move; Matter tells spacetime how to curve". I have a fairly good mental picture of how the first part works. However, I don't have much of an understanding of the second part. In particular, what I'd like to know is whether the equations describing how matter tells spacetime "could have been different." In other words, did Einstein choose the form of his equations to fit empirical observations, or are they the sort of thing that are worked out entirely from first principles and can't be changed at all without breaking the whole theory? Evidently there was some choice in choosing the equations, namely the cosmological constant - Einstein included it originally, then took it out, and now it seems like it might be there after all. But is adding a cosmological constant the only possible way in which Einstein's equations can be modified? Another way to ask this question is to ask what assumptions are required to derive general relativity. I know that deriving special relativity really only requires the principle that the laws of physics (including Maxwell's equations) are the same in all intertial reference frames, and I know that the first part of Wheeler's quote comes from the principle of equivalence between gravity and acceleration. But what, if any, additional assumptions are required in order to determine the way in which matter curves spacetime?


title: Definition of the complex wavenumber

text: My syllabus of electromagnetism defines the complex wavenumber as: $$k = \omega\sqrt{\epsilon\mu}$$ with $\epsilon$ the complex permittivity and $\mu$ the complex permeability. Thus $\epsilon$ and $\mu$ are complex numbers, which makes the square root ambiguous. They state that this issue can be solved, by defining $k$ as a complex number with a negative imaginary part and a positive real part: $$k = \alpha-i\beta$$ with $\alpha \ge 0$ and $\beta \ge 0$. I don't see though how this can be defined this way, because it seems to me that it is possible that $\epsilon\mu$ has no square roots in the fourth quadrant. I hope someone can clarify this.


title: How can a body be transparent?

text: How can a body be transparent? I need the theoretical explanation for the same. Has anybody succeeded in doing that practically?


title: Bell states entanglement

text: I'm trying to learn about the Bell state $\frac{1}{\sqrt{2}}|00\rangle+\frac{1}{\sqrt{2}}|11\rangle$. Question 10.1 in Algorithms asks us to show that this cannot be decomposed into the tensor product of two single qubits' states. It seems to me however that this can be decomposed while still obeying the basic rules. Wolfram Alpha lists some solutions. What am I doing wrong?


title: Formation of the overlap in metal electron bands

text: I understand that metals have overlapping of valence and conduction bands. But is this because there exists a partial conduction band within the top part of a metal valence band, or because the conduction band exists, but periodically in the valence region? Is this enabled by the formation of 'covalent' molecules in the metal structure matrix?


title: Can "state" be considered a 5th dimension?

text: I searched for an answer to this question on Google but just found articles that mention either string theory or a 5th dimension in passing (such as Maxwell equations as they relate to Riemann curvature tensor.) I stumped myself while driving to school today thinking about this... We can explain an objects position in the universe by describing its spacial and time locations, such as _at 2nd and 3rd street on the fifth floor at 10:00am_ But is that enough to fully describe an object when you take into account Schrödinger's cat? For example, what if, Schrödinger's cat is the object that is at that location and time. It seems like you would have to have a 5th _state_ dimension to fully explain its position, as in _at 2nd and 3rd street on the fifth floor at 10:00am with a probability of .5_ It seems like if we had _state_ as an additional dimension it would help explain things like quantum entanglement as the particles could be moving away from each other in space-time but standing still in state. Am I merely relating things that have no business being related, or is there a connection between state and dimension?


title: Accessible microstates of harmonic oscillator in microcanonical enemble

text: While reading up on statistical physics, I am going through the calculation of the partition function of the harmonic oscillator in the microcanonical ensemble. The result for the partition function is: $\Omega(E) = \frac{E_o}{\hbar \omega}$, i.e. all one-dimensional harmonic oscillators of the same frequency $\omega$ have the same number of accessible microstates. Consequently, as $\omega$ grows, $\Omega(E)$ goes to zero. This is somewhat counter-intuitive to me. I would have expected that an oscillator with a higher frequency $\omega$ would have more accessible microstates than one with a lower frequency. What is wrong with my intuition? Note that I am going through the classical calculation, the partition for the harmonic oscillator is then given as: $\Omega(E) = \frac{E_0}{h}\int dp \int dx \delta \left( \frac{p^2}{2m} + \frac{1}{2}k x^2 - E\right) $


title: Quantum oscillator, position mean value problem

text: A quantum harmonic oscillator of mass $m$ and frequency $\omega$ is at time $t=0$ in the state: $$ \left|\psi(t)\right> = \sum_{n=N-\Delta N}^{N+\Delta N}\left|n\right>\frac{1}{\sqrt{2\Delta N +1}}, $$ where $\left|n\right>$ is the $n$-th eigenket of the number operator and $N\gg\Delta N \gg1$. Show that at time $t>0$: $$ \left<\psi(t)\right|\hat{q}\left|\psi(t)\right> \simeq \sqrt{\frac{2 \hbar N}{m \omega}}\sin \omega t. $$ My try, as follows. First we write the time dependent ket, using the time evolution operator: $$ \left|\psi(t)\right> = e^{-i\hat{H}t/\hbar}\left|\psi(0)\right> = \sum_{n=N-\Delta N}^{N+\Delta N}\left|n\right>e^{-i\omega t(1+n/2)}\frac{1}{\sqrt{2\Delta N +1}}. $$ Then we write the expectation value: $$ \left<\psi(t)\right|\hat{q}\left|\psi(t)\right> = \left(\sum_{n'=N-\Delta N}^{N+\Delta N}\left<n'\right|e^{+i\omega t(1+n'/2)}\frac{1}{\sqrt{2\Delta N +1}}\right)\hat{q}\left(\sum_{n=N-\Delta N}^{N+\Delta N}\left|n\right>e^{-i\omega t(1+n/2)}\frac{1}{\sqrt{2\Delta N +1}}\right)=\\\ \frac{1}{2\Delta N+1}\sum_{n, n'=N-\Delta N}^{N+\Delta N}e^{i\omega t(n'-n)}\left<n'\right|\hat{q}\left|n\right>. $$ Now, using the creation $\hat{a}^\dagger$ and annihilation $\hat{a}$ operators we have: $$ \hat{q} = \sqrt{\frac{\hbar}{2m\omega}}(\hat{a}+\hat{a}^\dagger), $$ therefore: $$ \left<n'\right|\hat{q}\left|n\right> =\sqrt{\frac{\hbar}{2m\omega}}\left( \left<n'\right|\hat{a}\left|n\right> \+ \left<n'\right|\hat{a}^\dagger\left|n\right>\right)=\sqrt{\frac{\hbar}{2m\omega}}\left(\delta_{n', n-1}\sqrt{n}+\delta_{n',n+1}\sqrt{n+1}\right). $$ Substituting back into our expression yields: $$ \frac{1}{2\Delta N+1}\sqrt{\frac{\hbar}{2m\omega}}\sum_{n, n'=N-\Delta N}^{N+\Delta N}e^{i\omega t(n'-n)}\left(\delta_{n', n-1}\sqrt{n}+\delta_{n',n+1}\sqrt{n+1}\right) = \\\ \frac{1}{2\Delta N+1}\sqrt{\frac{\hbar}{2m\omega}}\sum_{n=N-\Delta N}^{N+\Delta N}\left(e^{-i\omega t}\sqrt{n}+e^{i\omega t}\sqrt{n+1}\right)=f(t). $$ By bringing the derivative under the sum we see that: $$ \frac{d^2f(t)}{dt^2}+\omega^2f(t)=0 $$ which means: $f(t) = A \cos \omega t + B \sin \omega t.$ All we have to do now is finding the coefficients. We know that: $$ f(0) = A = \frac{1}{2\Delta N+1}\sqrt{\frac{\hbar}{2m\omega}}\sum_{n=N-\Delta N}^{N+\Delta N}\left(\sqrt{n}+\sqrt{n+1}\right)\simeq\\\ \frac{1}{2\Delta N+1}\sqrt{\frac{\hbar}{2m\omega}} \int_{N-\Delta N}^{N+\Delta N} dn \left(\sqrt{n}+\sqrt{n+1}\right)=\\\ \frac{1}{2\Delta N+1}\sqrt{\frac{\hbar}{2m\omega}} 2\sqrt{N}\left(2\Delta N +\frac{1}{2} - \frac{1}{2}\right), $$ where in the last passage we did the trivial integration and then used a first order Taylor expansion. Similarly, B yields zero at first order in $\Delta N /N$. So we have: $$ \left<\psi(t)\right|\hat{q}\left|\psi(t)\right> \simeq \sqrt{\frac{2 \hbar N}{m \omega}}\cos \omega t \frac{2\Delta N +1/2 - 1/2}{2\Delta N +1/2 + 1/2} $$ which is not quite what I was supposed to find. Thanks in advance.


title: Why do rocket engines have a throat?

text: Diagrams of rocket engines like this one, ![rocket engine](http://i.stack.imgur.com/EZWGCm.png) (source) always seem to show a combustion chamber with a throat, followed by a nozzle. Why is there a throat? Wouldn't the thrust be the same if the whole engine was a U-shaped combustion chamber with a nozzle?


title: What is the meaning of a constant magnetic scalar potential?

text: Let a spherical shell of inner radius $a$ and outer radius $b$ have a uniform magnetization $\mathbf{M}=M\,\hat{\mathbf{z}}$, $\hskip2in$ ![Transversal view of spherical shell with uniform magnetization](http://i.stack.imgur.com/AFpa4.png) I've found that the magnetic scalar potential $\varphi^\star$ is given by \begin{align}\varphi^\star(r<a)&=\text{constant}\\\ \varphi^\star(a\leq{r}\leq{b})&=\frac{M}{3}\left(r-\frac{a^3}{r^2}\right)\cos\theta\\\ \varphi^\star(r>b)&=\frac{M}{3r^2}\left(b^3-a^3\right)\cos\theta\end{align} where $\theta$ is the polar angle. I guess my calculations are ok because when $a\to0$, the results for a sphere with uniform magnetization $\mathbf{M}=M\,\hat{\mathbf{z}}$ are recovered. Anyhow **I got confused by the constant potential inside the shell, how can I interpret it?** If $\mathbf{B}=\mu_0(\mathbf{H}+\mathbf{M})$ and $\mathbf{H}=-\nabla\varphi^\star$, then there would be a uniform magnetic field inside the shell and still if I put a charged particle there, also a force upon it, so **how is this potential different from any non-constant potential $\varphi^\star\propto{z}$, as the one inside the complete sphere (i.e. when $a\to0$)?**


title: Short term track of a hand held sensor (gyro, accelerometer) data with rough precision

text: I have acceleration and attitude data for the movement of a device in 3d space. what transformations, before numerical integration, are required, in order to produce a 3D track of the the movement in the earth's frame of reference. I know about the precsion and error issues. And I am not looking for commercial or mil spec accuracy. I want to try. I have seen the transformations in various papers. Can anyone suggest links or code that could accelerate my efforts?


title: heat energy and resistance of an elctric circuit

text: Very often it is said that "heat is developed" in a resistance when there is an electric current in it. But as far as I know heat is defined as a form of energy being transferred due to difference in temperature between two points. Now I am confused, is the statement inside the quotes technically correct?


title: Dipole moment dilemmas

text: Dipole moment for a couple of charges, say q and -q, separated by a distance 'd' is given by 'qd' But what is for dissimilar magnitudes,say (q and 2q) or (q and -2q)? And are dipole moment defined only for static charges or are they defined for moving/accelarated charges too..?


title: Conformal group in two dimensions

text: How can one show in a group-theoretical way that each of SO(d,2) and SO(d+1,1) is isomorphic to two-copies of Virasoro algebra for d=2?


title: Energy time complementarity from unitary evolution

text: I am looking for a well posed experimental situation that illustrates energy time complementarity. I know of Einsteins box, which is discussed quite nicely in Bohr's article Discussions with Einstein on Epistemological Problems in Atomic Physics. He used time dilation due to gravity, I am looking for examples that do not involve gravitational physics. Bohr in his article never applies unitary evolution $e^{-iHt}$ directly to the problem. This is analogous to $e^{ipx}$ which is at the heart of momentum position uncertainty . It would be nice to see Energy time Complementarity directly from $e^{-iHt}$. What does energy time complementarity imply for energy conservation? Does such an experiment exist? If not, is there some fundamental impedance preventing it from being done in a gravity-free context? Please take care to pose the experimental situation completely, as these are subtle issues. I would prefer examples that do not involve QFT, as it would unnecessarily distract us from the main question.