title: Why do we set $x^0 = ct$ instead of $x^0 = t$?

text: When we deal with Special Relativity and we start considering spacetime instead of space and time each at once, we usually see books saying that we consider a space with four coordinate $x^\alpha$ with $x^0 = ct$. We also consider this manifold to be $\mathbb{R}^4$ and give to it the metric tensor $g = \operatorname{diag}(-1,1,1,1)$. Now, why $x^0 = ct$ and not just $x^0 = t$? One possible answer might be "so that all coordinates are measured with the same units", but isn't there any deeper meaning?


title: Lie algebra and Lie group about quantum harmonic oscillator

text: We know that in the quantum harmonic oscillator $H=a^\dagger a$, $a^\dagger$, $a$, $1$ will span a Lie algebra, where $a, a^\dagger$ are the annihilation and creation operators, and $H$ is the Hamiltonian operator. $$[H,a^\dagger\ ]= a^\dagger$$ $$[H,a]=-a$$ $$[a,a^\dagger]=1$$ So these four operators, $H=a^\dagger a$, $a^\dagger$, $a$, $1$, can span a Lie algebra, because the commutator satisfies closure and Jacobi's identity. We know that for any Lie algebra $\mathscr{G}$ there exists only one Lie group $G$ up to a difference in the topology, whose Lie algebra is $\mathscr{G}$. So what is this Lie group whose Lie algebra is spaned by $\\{H=a^\dagger a , a^\dagger ,a ,1\\}$ ?


title: Position and potential Energy

text: Why are the position and potential energy of a particle able to be measured precisely in Quantum Mechanics? I mean why do they commute with each other?


Referencing Stephen Hawking's recent paper _Information Preservation and Weather Forecasting for Black Holes_ and this question. I understand concept of _holding_ the information on the apparent horizon of a black hole for later release in the form of garbled radiation, but how is it that this differs ( if at all ) from the Hawking Radiation generated by the Unruh effect near the horizon? My, possibly flawed, train of thought is that the aforementioned black hole sucks in everything in matter in it's immediate surroundings, and due to the Feynman diagram that corresponds to particle-antiparticle pairs spontaneously popping into existence with one particle on the other side of the horizon, the energy from the matter sucked in, is imparted to which ever particle is outside of the event horizon to repay the energy debt to the Universe.


title: Evidence for black hole event horizons

text: I know that there's a lot of evidence for extremely compact bodies. But is there any observation from which we can infer the existence of an actual _horizon_? Even if we are able to someday resolve directly the "shadow" of a possible black hole, isn't it still possible that it's just a highly compact body that's just gravitationally redshifting any radiation that it emits to beyond the detection capability of our telescopes? What would be indisputable evidence for a causal horizon?


title: What is the meaning of the word "Principle" in Physics?

text: What is the meaning of the word principle in Physics? For example in the "action principle". Is it an action law, an action equation, or an unproved assumption? (I have an idea what an action is). What confuses me is when I find the expression as in: "...an action principle is defined..." when a Lagrangian is constructed for a system.


title: Laws of Atomic Theory - how is this possible?

text: Not sure if this is the right place to post, but how is it possible to have laws of a theory? A theory is not able to be a law, since it's just an explanation that can always be disproven. So how is there a law of a theory?


title: Question about physical principles

text: How are principles created i.e. how is it decided that something qualifies as principle? What is the difference between a principle, a law and a theory? Were there any principles that were proved to be wrong or are principles always right? For example, the reason why Einstein proposed that the speed of light is constant is because the Principle of Relativity would be violated otherwise. Are physical principles comparable to mathematical axioms? Is there a list of physical principles? I couldn't find any on google.


title: Why Is String Theory Called A Theory

text: > **Possible Duplicate:**   >  Laws of Atomic Theory - how is this possible? Generally in science, a theory is effectively a fact. The **theory** of evolution is not a guess, or a hypothesis. It's currently understood to be a fact, backed up by evidence from numerous scientific fields spanning many decades. String **theory** on the other hand seems much more contentious, much less settled in the scientific community. Why isn't it called the string hypothesis?


title: Space-time filling D-branes in Type I superstring theory

text: It's known that gauge group for type-I superstring theory should be SO(32) for chiral anomalies to cancel. This is realized by presense of 16 space-time filling D9-branes, together with 16 space-time filling O9-planes with opposite charge: D9-branes bring U(16) gauge symmetry and due to orientifold planes the symmetry gets enhanced to SO(32). But what about other Dp-branes which can exist in type-I theory: wouldn't they bring anomaly - because open strings can end on those branes and create gauginos from gauge group different from required SO(32) - and will that anomaly cancel eventually (people make physical constrctions with other Dp-branes e.g. while studying BH in superstring theory)? Or in any picture we are always supposed to be able to perform enough T-duality transformations and get our 16 D9-branes and 16 O9-planes, but there will be additional branes. Probably the answer to my question is that all other possible branes in that picture will be part of 16 space-time filling branes, and therefore gauge YM group will be then SO(32) anyway and other gauge groups will be just subgroups of it - hence in the whole system anomalies will cancel, is it true?


title: What is tension in string theory?

text: One often hears the words "string tension" in string theory. But what does it really mean? In ordinary physics "tension" in an ordinary classical string arises from the fact that there are elasticity in the string material which is a consequence of the molecular interaction (which is electromagnetic in nature). But string theory, being the most fundamental framework to ask questions about physics (as claimed by the string theorists) can not take such elasticity for granted from the start. So my question is, what does "tension" mean in the context of string theory? Perhaps this question is foolish but please don't ignore it.


title: How small of a depletion signal can the best modern mass spectrometers detect?

text: So I know the answer to this question varies widely across mass spec techniques, masses, and, of course, budgets, but my question is about the _best_ case scenario for all of these variables (although I'm not really looking at proteins or anything of that size). I realize that the concept of depletion spectroscopy relies heavily on very sensitive mass spectrometers, and I'm curious about the lower limits of signal detection for such methods. Thus, I'm interested in the sensitivity, not resolution ("mass resolving power"), which seems to be a statistic that is difficult to find for most of the systems that I've looked at. Some related follow-up questions are what mass spectrometry methods can and cannot be used with depletion spectroscopy and what methods/manufacturers are the gold standard in the field of mass spec? Thanks in advance for your answers.


title: Electrical eddy current visualization or simulation

text: Eddy currents are induced in a metal plate when it experiences a changing magnetic flux. Is there a realistic visualization or simulation of eddy currents available? The only picture I found, on wikipedia, is a cartoon schematic.


title: What exactly are right movers and left movers in string theory?

text: In string theory , What exactly are right movers and left movers ?Are they waves propagating along the string to the right and to the left respectively ? Can some-one please show me how to derive in details the general solution of the equation of motion for the string?


title: Capillary engine is impossible?

text: I read some articles saying that it is not obvious (or maybe impossible) to generate energy using capillary action. On the other hand in my understanding trees uses capillary action to extract water from the soil, but I guess the "trick" there is that the sun actually extract the water from the plants making it evaporate.   * Has it been proved impossible to generate energy with capillary action, or is it just believed to be impossible?   * Is there any device that mimics plant action?


title: Time-ordered calculation for equal time

text: I have a question about how to calculate the following expectation value: $$\langle0|\mathcal{T}\\{{a^{\dagger}}(0,0) a(0,0)\\}|0\rangle$$ where $|0\rangle$ is the ground state and $a^{\dagger}(x,t)$ is the creation operator which creates a particle at position $x$ and time $t$. Is the answer just $0$ since time-order operator does nothing in this case?


title: Spontaneous symmetry breaking and 't Hooft and Polyakov monopoles

text: What is spontaneous symmetry breaking from a classical point of view. Could you give some examples, using classical systems.I am studying about the 't Hooft and Polyakov magnetic monopoles solutions, which says that the SU(2) symmetry is spontaneously broken down to U(1) symmetry. What does this mean exactly? Also, why do we consider a mexican hat potential in the yang-mills- higgs lagrangian, whose EOM give us the solutions to the monopoles, using the Euler-Lagrange equation. I tried reading about this from the internet, but most of the sources have reference to quantum systems ( I havent done QM and QFT), and I haven't been able to understand symmetry breaking from the few classical examples that I found.


title: Using Einstein's Relativity: Who is younger?

text: Suppose we have a person **A** and a person **B**. Person B travels very close to speed of light and never returns. He's constant in speed. Then, we can say two things:   1. B is younger than A.   2. A is younger than B (since we can consider B's reference as inertial). Who is correct between the two?


title: Can any solid of low heat capacity release or gain energy slower than another solid of higher heat capacity?

text: If our sense of contact temperature is influenced by the thermal conductivity of the material: _Can any solid material with a low heat capacity exist that feels closer to human body temperature than another solid material with a higher heat capacity; where both materials were previously kept in either a mundane oven or freezer for a sustained period?_ * * * I start thinking about whether heat capacity will always be symmetrical for gaining and losing energy and then I realise I've moved beyond my level of Physics knowledge.


title: Schrödinger Equation: Eigenmomentum?

text: I'm confused by my books treatment of the Schrödinger equation. In steado f listing my questions at the end of my post, I'll add them as questions in parentheses after the line in question. For a free particle: $$i \hbar \left| \dot{\Psi} \right \rangle = H\left| \Psi \right \rangle = \frac{P^2}{2m}\left| \Psi \right \rangle$$ (where did the potential go?) The normal mode solutions are of the form: $\left| \Psi \right \rangle = \left| E \right \rangle e^{-iEt/ \hbar}$ Feeding this into the equation above (the schrödinger equation written above), we get the time-independent equation for $\left| E \right \rangle$ : $$H \left| E \right \rangle = \frac{P^2}{2m}\left| E \right \rangle = E \left| E \right \rangle$$ (this follows from the eigen-equation, where the eigenvalue must be equal to $E$?) This problem can be solved without going into any basis. First note that any eigenstateof $P$ is also an eigenstate of $P^2$. So we feed the trial solution for $\left| p \right \rangle$ into the equation (the equation directly above). $$\frac{P^2}{2m}\left| p \right \rangle = E\left| p \right \rangle $$ (why are we all of a sudden talking about the ket $p$? Which means that $\left| p \right \rangle = \pm \sqrt{2mE}$ This gives us two eigenkets of $E$, which span an eigenspace. The next steps are to set up the propagator (which I have questions about, but understanding the previous steps may help me clear them up myself).


title: What will be different if the spring is not massless?

text: In almost all text, it uses the massless spring as example to illustrate the idea of elastic potential energy. I wonder what's really being changed if we consider the mass of the spring? I saw a problem on a text regarding the non- massless spring. It is said that the stretched spring (don't know the length) has uniform density and mass $M$, the elastic constant k is also known. If we hold it vertical so it stretches naturally due to the gravity, what's the elastic potential will be. I am trying to understand this from physical point of view as follows but I don't know if it is correct or not since there might be some knowledge beyond undergraduate course I need to solve the problem 1) I think the spring is actually has the gravity acting on it's center of mass, so if we choose the (gravitational) potential reference to be zero (at where the hanging point), the gravitational potential is $Mgh$ where $h$ is the where the center of mass is . 2) From 1), can we say that the elastic potential energy is actually the same as the gravitational potential energy, so $$\frac{k}{2}(\Delta l)^2 = Mgh$$ So if we figure out $h$, we know the elastic potential energy. But the problem how to find the center of mass. I bring this to some senior students, some of them said the center of mass will not be changed, so the center of mass when the spring is not stretched and when it is naturally stretched will be the same, that's is to say, when it is not stretched, the center of mass is $l_0/2$ so $h=l_0/2$. Well, frankly, I don't see the point. Someone show me the math to calculate the center of mass, which is an integral $$ h = l_c = \frac{1}{M}\int_0^L \frac{M}{l_0}xdx $$ there he said $L$ is the fully-extended length and $l_0$ is the unstretched length, I don't know how to do the math but I plug that into mathematica and it gives $$ h = l_c = \frac{L^2}{2l_0} $$ but if this is correct, we need to know the fully-extended length and original length to figure the center of mass. I don't think we get that information. So do we really need other knowledge to find the elastic potential energy for non-massless spring?


title: How many sigma did the discovery of the W boson have?

text: When the W boson was discovered in the 1980s, nobody spoke of sigmas. How many sigmas was it at that time?


title: Is space unending?

text: Is the space unending, i.e it has no boundaries? If yes, how can a thing exist which is non-ending? Its impossible for me to imagine something like that. Secondly, if its not and has boundaries then it must be inside some other thing which is either non-ending or itself contained in another thing. Yet, again same problem. If it is also inside some other thing, and the other thing is also inside other thing, then this series will never end. Imagining about this scenario makes me insane.


title: Is it possible to send a single photon from a distant planet (say Mars) and detect its arrival at a site on Earth?

text: My question is specifically whether there exists a technique by which a **_single_** photon can be "tagged" or "encoded" in such a way that it can traverse our atmosphere and arrive at some sort of detector along with trillions of others and be singled out from the other photons. Would this be possible using quantum entanglement protocol? Or maybe precise timimg (knowing when the photon will arrive at the detctor based on knowing when it leaves and the distance between)? Thanks


title: Does regular sky clouds stops the sun's ultraviolet rays?

text: I have an argue with an ex-Doctor of Medicine about the amount of ultraviolet light reaching us under the clear day sun and under the 100% cloudy sky. To what extent we can say that the sky clouds stops ultraviolet light from the sun?


title: Why are the noble metals inert?

text: I am wondering: The noble 'gases' are inert because they have closed shells and don't want to give that up. But the noble metals, such as Copper, Silver, Rhodium, Gold, don't seem to have this. For example, Rhodium has electron configuration $4d^{8}\, 5s^1$. To me, this looks very similar to the alkali metals, which we all know are 'very' reactive. So why doesn't Rhodium happily give up its s-electron? Same with Silver, $4d^{10}\, 5s^1$. Why not give up the s electron?


title: Inhomogeneties in gravitational acceleration at the surface of the Earth.

text: I am looking for data sets which clock the time and precise location of free falling bodies in a vacuum. I have been told that the acceleration of gravity is not absolutely 100% uniform, and I would like to examine the data sets as to this question, particularly as to minute discrepancies. The more the better. Any help in locating this raw data would be very appreciated.


title: What physical properties of silver would contribute to the sound of a musical instrument?

text: Sorry if this is off-topic. A question was recently asked on Musical Practice & Performance, asking what physical properties of silver would contribute to the sound of percussion instruments. It is likely to be off-topic there; I wondered if any of you physicists/material scientists would be able to offer a humble musician information on this site. It would also be interesting to know what contribution solid-silver or silver- plated components would make to brass/woodwind instruments, and what properties of silver are responsible for this.


title: Acer saccharum (Sugar Maple) Sap & Freezing

text: This may or may not be the appropriate scientific discipline to ask this question of, but I'll give it a shot. I'll happily invite chemists to chime in as well. It's time to tap sugar maples in the North East USA once again. What we essentially pull from the trees via taps is a sugar solution that, using reverse osmosis and evaporation, produces maple syrup. But there's a old tradition that's caused me a bit of concern. When buckets are hung (a dying method of collecting, usually big operations use tubing), sap is collected as the day warms up. At night, and especially on very cold nights, large portions of the sap freeze. I've noticed a lot of older sap producers toss out the ice, sometimes a substantial amount, saying that what is left is more concentrated sugar-water, and the ice is pure water. But I'm not sure if I believe that. First of all, most sap is a 2% to 6% sugar solution, meaning that, at 2%, you'll need roughly 43 gallons of sap to produce 1 gallon of syrup. Syrup is defined as > or = 66% sugar (Sucrose). And I've seen sap run through reverse osmosis (which is then about 10% sugar), freezing up after an overnight below zero. With that bloated introduction, my questions are as follows: Assuming you have 1 liter of pure water, vs 2% sap, vs 10% sap (solution I assume), at what temperatures will each solution start to freeze. Secondly, when any of these solutions freeze, is it only pure water that is freezing initially? Or is the ice actually a mix of sucrose and pure water? And if it is separating pure water from sucrose, forming a more concentrated solution, at what temperature or time (again, assuming 1 liter and a constant atmospheric pressure) will the more concentrated solution start freezing? Finally, many pails and the sap have lots of particles in it. This can range from bugs to wood chips to bits of dust, dirt, and corrosion. Although it might seem a separate topic, this may also influence freezing rates and should be taken into consideration when trying to answer this question. The point is this: We don't want to throw away the valuable sucrose the tree is providing, but we want to be more efficient in removing water from the solution to form syrup. Is tossing out ice that forms in the buckets overnight helping with that efficiency, or are we actually throwing out valuable sucrose that could be made into syrup?


title: About electric current analogy

text: my teacher gave me this analogy to the electric current , the wire is like a pearl necklace where the pearls can move, the current or the movement of electrons is like putting your fingers between 2 pearls separating them then you start taking one pearl from the right of your finger and push itto the left of your finger starting making the pearls to move, is this analogy right?


title: Instantaneous displacement of a traveling sine wave with an undisturbed starting condition

text: The equation for the instantaneous displacement of a traveling wave at time $t$, and at any position $x$ along the wave is: $$y(x,t) = A_{max}\sin(kx +\omega t + \phi)$$ ![enter image description here](http://i.stack.imgur.com/nxG3P.gif) But this is only valid when the traveling wave has already formed (i.e in steady state). If the wave at time $t=0$ has displacement $0$ at every position $x$, what will the equation be then? ![enter image description here](http://i.stack.imgur.com/UaY30.gif) And what is the equation when there is a full or partial reflection? For a standing wave in steady state the equation is: $$y(x,t)=2A_{max}\cos(\omega t)\sin(kx)$$ But what is the equation that also describes the transient period, as shown in the picture below? ![enter image description here](http://i.stack.imgur.com/cc2Lc.gif)


What do Allowed and Forbidden transitions tell us about the properties of a photon? Allowed transitions have the change in angular momentum $\ell=1$, all the others are forbidden. But what does it tell us?


title: A question about quantum measurement and associating a linear self adjoint operator to it

text: I have a question about the concept of measurement and observable in quantum mechanics. I'd like to fist explain my understanding of it and then ask the question. First we have a system and its state function $\psi(x)$ and naturally it means there are two mutually incompatible observables $x$ and $\omega$ (momentum). Now we'd like to set up an experimental setup and verify whether this state function actually desribes the given system. Consider an example of tossing an unbiased coin. We want to verify whether this coin is unbiased or not. There are two mutually exclusive outputs heads and tails. We repeat this experiment infinite number of times (actually only very high number of times) and record how many heads and tails resulted there by we know the probability of heads and tails. Similarly we set up the quantum system, and perform the experiment repeatedly and every time we note the value of $x$ (position of particle) and there by we get the probability distribution of $x$ and hence can verify the validity of the state function $\psi(x)$. But this is a very naive way of doing the experiment as we have to set up particle sensors every where in space (all $x$) and it they should be able to detect particles at any velocity. This kind of experiment is not easy to set up. What if we already have some equipment and experimental set up to verify a certain type of state functions. We may use them to our advantage. Suppose we have measuring equipment which can directly verify the occurence of certain state functions $f_1$,$f_2$,... Assume the experiment gives out to a system into different possible states $f_1$,$f_2$,$f_3$,... which are mutually exclusive that is the systems cannot jump to two state $f_1$ and $f_2$ simultaneously. which means if system jumps to $f_1$ then it will not jump to any state $f_k$, $k \ne 1$. Let us assume that the probability of jumping from a state $psi_1$ to a state $\psi_2$ is given by the inner product $<\psi_1|\psi_2>$. Now that the states $f_1$,$f_2$... are mutually exclusive outcomes, and probability of transitioning from one state to other is given by their inner product, the set of states $f_1$,$f_2$,... form a orthogonal set. Since the probabilty of transitioning from one state to another is given by the inner product, these states form an orthogonal set and $\psi$ can be expressed as a linear combination of these states.( Lets us assume this set is a complete orthonormal set so that we can generalize it for any $\psi(x)$). Now what we are actually doing is we have an experimental setup to detect certain set of states which form a complete orthogonal basis of the Hilbert space. We can repeat the experiment to find the probability of occurrences of these states. We have chosen the experimental set up for detecting $f_1$,$f_2$... as we already have such an equipment, means we are doing a certain type of measurement defined by the set $f_1$,$f_2$,... Which means the measurement is defined by choice of complete orthogonal set $f_1$,$f_2$,... and evry such set defines a linear, self adjoint operator $L$ on hilbert space, we can associate/define every measurement by a linear self adjoint operator. Hence measurement = linear self adjoint operator. It can be mathematically easily verified that the set of states $f_1$,$f_2$,... are eigen vectors of the linear operator, and since we choose inner product as the probability of jumping from state to another state, it can be easily seen that the eigen values associated with these vectors give the probability of their occurence. **Question** Why do we always need to choose a a orthogonal set which is **complete**? I mean Why we need to choose a measurement defined by a linear operator. What I say is can't we choose a set of orthogonal set which is not complete but it be used to express the our state vector as a linear combination. I mean why cant we choose a set of orthonormal vectors which is not complete, but $\psi$ lies in its span. After all thats wht we want. We can choose this set based on the $\psi$ we have and also our equipment. We just don't want to generalize a measurement to every $\psi$, but we choose measurement based on $\psi$ we have. **(Added after answer by SMeznaric)** Thanks for the illustrous answer. My question is not against linearity or orthogonality, but regarding why formalize and give importance to a universal orthogonal basis. I mean why should we even talk about them. On the contrary Lets say we decide the orthogonal basis purely on the ψ(x) at hand. Unless you give ψ(x) we will not talk about anything. We use linear orthogonal basis states, but our measurement/basis will totally be adapted to the given ψ(x) and we will not talk anything unless you give the ψ(x) to be analyzed. ![enter image description here](http://i.stack.imgur.com/D3bb9.jpg) Consider this example of a state function $\psi(x)$ of an electron. Now it tells me that the best way to verify this is to put a electron detector and a filter (allows only electrons of 16 units/s) covering the region of (0,1) and place a similar one but with a filter of 32 units/s covering (1,2), now you are done. But this is based on common sense, now we need a mathematical model for this, which chooses the basis based on the state function, and yet make very useful in practical terms. (suitability to our equipment).


title: Why is Standard Model + Loop Quantum Gravity usually not listed as a theory of everything

text: I have often seeen statements on physics.SE such as, > The only consistent theory of everything which we know of to date (2013) is > string theory. Why exactly is this so? Adding the Loop Quantum Gravity Lagrangian Density (the Einstein-Hilbert-Palatini-Ashtekar lagrangian density) to the Standard Model Lagrnagian Density should be able to describe all the interactions and fermions, in my opinion. Maybe it isn't as elegant as string theory since it doesn't really unify all the forces/interactions and fermions but it is still a complet description, right? Because once the Lagrangian Densities are added, one obtains the following "Complete Lagrangian Density": $${{{\cal L}}_{\operatorname{complete}}} = - \frac{1}{4}{H^{\mu \nu \rho }}{H_{\mu \nu \rho }} + i\hbar {c_0}\bar \psi \not \nabla \psi + {c_0}\bar \psi \phi \psi + \operatorname{h.c.} + {\left\| {\not \nabla \phi } \right\|^2} - U\left( \phi \right){\rm{ }}+\Re \left( {\frac{1}{{4\kappa }}\mbox{}^ \pm\Sigma _{IJ}^\mu {{\rm{ }}^ \pm }F_{IJ}^\mu} \right) $$


title: Taylor series expansion of $\ln$ and $\cosh$ in distance fallen in time $t$ equation

text: I want to find the Taylor expansion of $y=\frac {V_t^2}{g} \ln(\cosh(\frac{gt}{V_t}))$ I have tried using the fact $\cosh x= \frac {e^x}{2}$ for large t, which works, I just need help on small values for t. Using maple17 for small t, I was able to see it expands to $\frac{1}{2}gt^2 - \frac{1}{12}\frac{g^3}{V_t^2}t^4 - o(t^6)$ but I cannot see how it got to this.


title: Am I right in saying that $gmm/r$ is motion due to potential energy?

text: I was watching a video about the swarzchild radius and it said that potential energy is $gmm/r$. This cannot be right though because potential energy goes up with distance not down. I'm assuming he meant motion due to potential energy. Also what is the intuition behind the equation $gmm/r$?


title: Electromagnetic Theory

text: As, using the fact that E*A stays constant for a point charge ( E= electric field, A=area of sphere centered on charge touching the point where we have to measure E) , we can prove that E falls by 1/r^2 . Is there any basic method for the Magnetic field?


title: Are nodes and orbitals in atoms simply probability distribution clouds or are they of any physical relevance?

text: I fail to understand what the electron clouds actually signify. Such as the $p$ orbitals, which have a dumbbell like shape. Now I am aware that they aren't actual trajectories of electrons, but what is their need? Does it indicate the highest probability of finding an electron? And also, concerning the Zeeman effect, which I believe is the splitting of spectral lines in a static magnetic field (correct me if I am wrong), we have been told that this occurs due to separation of orbitals into their components. Assuming the $p$ orbital is affected by a magnetic field, it would split into $p_x, p_y$ and $p_z$ components, what does this mean, and how do these probabilities (if that assumption is right) actually exist in its essence? I would really appreciate some insight into this relatively new topic. And one last thing concerning, the Radial probability density and Radial probability distribution curves, what does the graph of $4\pi r^2R^2$ vs. $r$ plots attempt to impart? The local maximas represent the radius of the nodes in an atom, but how is that specific graph derived?


title: How do I find the energy of this Hamiltonian?

text: So, I have a (confusing) Hamiltonian: $$H = \int \mathrm{d}k\,\omega_{k} a^{\dagger}\bigl(\vec{k}\bigr)a\bigl(\vec{k}\bigr)$$ where $\omega_{k} = \sqrt{k^{2} + m^{2}}$ and the measure is $\mathrm{d}k = \frac{\mathrm{d}^{3}k}{(2 \pi)^{3} 2 \omega_{k}}$. I want to find the energy of a single particle state $\lvert k\rangle = a^{\dagger}(k) \lvert 0\rangle$ and ask the energy of a multi-particle state, but I am not sure how to deal with that starting Hamiltonian. A lot of the work for this problem (getting the Hamiltonian into this form) was already done, but I am not sure how to make the final jump to the energies. This style Hamiltonian doesn't seem conducive to such a solution...


title: Using angular momentum in complex coordinates

text: So given the angular momentum operator: $$L_{z} = - ih\biggl(x \frac{\mathrm{d}}{\mathrm{d}y} - y \frac{\mathrm{d}}{\mathrm{d}x}\biggr)$$ I know how to write these in terms of polar coordinates (obviously). But say I want to work in complex coordinates, such that: \begin{align} z &= x + i y & z^{*} &= x - iy \end{align} and \begin{align} \frac{\mathrm{d}}{\mathrm{d}z} &= \frac{1}{2} \biggl( \frac{\mathrm{d}}{\mathrm{d}x} - i \frac{\mathrm{d}}{\mathrm{d}y}\biggr) & \frac{\mathrm{d}}{\mathrm{d}z^{*}} = \frac{1}{2} \biggl( \frac{\mathrm{d}}{\mathrm{d}x} + i \frac{\mathrm{d}}{\mathrm{d}y}\biggr) \end{align} I know that this should be possible (right?) but no matter how I dot the complex terms, I keep getting $x \frac{\mathrm{d}}{\mathrm{d}x}$ and $y \frac{\mathrm{d}}{\mathrm{d}y}$ terms that don't cancel and shouldn't be there. My textbook (Pringe) uses these substitutions in some problems but doesn't explain very well HOW one goes about performing them. Is it actually possible to work in these coordinates? Or is this just a notation that no one uses?


title: Hearing sound around corners

text: I've been taught that diffraction is the reason why we can hear around corners, since the wavelength of sound is approximately 10 cm (smaller than a door or a window). I wonder how can this be correct, since I can hear even when a room is closed (walls are not completely rigid), althought the sound will be lower when the doors are closed. On the other hand, reflection of sound with other objects would also contribute. Which is the most important process when hearing indirectly? Is there any way to calculate the contribution of each process?


title: Stringy corrections of Einstein's vacuum field equations

text: From string theory, the vacuum field equations obtain correction of the order $O[\alpha'R]^n$ such that they can be written as $$ R_{\alpha\beta} -\frac{1}{2}g_{\alpha\beta}R + O[\alpha'R] = 0 $$ where $\alpha' = \frac{1}{2\pi T_s}$, when including just the first order term for example. What is the physical interpretation of these corrections, what do they look like more explicitely, and how can they be obtained?


title: Helicity dependence in loop diagrams

text: I am trying to evaluate a diagram that looks like ![Feynman loop diagram](http://i.stack.imgur.com/9lbyJ.png) The middle of the diagram is a fermion loop. I know that the coupling between the $Z^0$ and fermions depends on the fermions' helicities, so it would be nice to evaluate the diagram separately for each fermion helicity. When I try to evaluate the diagram, I clearly end up with an expression involving fermion propagators, and I don't see how those carry helicity information. How can I deal with this? To calculate the amplitude for each fermion helicity, can I just insert the usual projector $(1 \pm \gamma^5)/2$?


title: Ising spin vs Pauli spin matrices

text: Are Ising spins scalar or operators? I am not a condensed matter physicist hence having some confusion. I have learnt about Ising models from adiabatic quantum algorithm papers. For example this presentation or this paper encodes their quantum adiabatic algorithm for an Ising model. In an Ising model the spins are $\pm 1$. In his original adiabatic quantum computation paper, Farhi didn't mention about Ising models. Moreover he explicitly used Pauli's spin matrices and not the $\pm 1$ spins. Can we use $\pm$ spins and Pauli's matrices interchangeability at least in the context of adiabatic quantum algorithms?


title: Why isn't the Earth's core temperature the average of its surface temperatures?

text: Assuming that the earth is spherical, that its temperature is continuous, and that some other more or less realistic conditions hold, we might think that the Earth's core temperature should be about the average of its surface temperatures. This is not the case, as the core is hotter than all but a few spots on the surface. Can someone explain where the assumptions break down? Is it that the temperature is not static?


title: maths question on levers

text: > Someone holds a 2kg weight in their hand at a distance of 35cm from the > elbow (fulcrum) and the downward force (load) due to the weight is 20 > Newtons. Calculate the effort in Newtons that the muscles of the forearm > must apply at a distance of 5cm from the fulcrum to hold the weight up. Show > calculations. Anyone any ideas please, I kinda grasp the concept but am unsure of how to show workings


Previously, on Save us from swallowing baby universes, please!, I asked how we can be saved from false vacuum decay assuming we are living in a false vacuum. What if we are living in a stable vacuum after all, and string theory actually describes the world we live in? Are there any stable (not metastable) string theory compactifications with a positive cosmological constant?


title: Decomposition of two particle wavefunction into product of single-particle wavefunctions

text: Suppose you prepare a two-particle system such that $\Psi(\vec{r}_1,\vec{r}_2, t_0) = \Psi_1(\vec{r}_1, t_0)\Psi_2(\vec{r}_2, t_0)$. So then, initially $\Psi(\vec{r}_1,\vec{r}_2,t_0) - \Psi_1(\vec{r}_1, t_0)\Psi_2(\vec{r}_2, t_0) = 0$ But now you let the system evolve in time and $\Psi(\vec{r}_1,\vec{r}_2, t)$ can now no longer be decomposed into a product of two single-particle wavefunctions. What then does, physically, does $\Psi(\vec{r}_1,\vec{r}_2, t) - \Psi_1(\vec{r}_1, t)\Psi_2(\vec{r}_2, t)$ represent?


title: Entropy increase and end of the universe

text: While taking thermodynamics our chemistry teacher told us that entropy is increasing in day by day (as per second law of thermodynamics), and when it reaches its maximum the end of the world will occur. I didn't see such an argument in any of the science books, is there any probablity for this to be true?


title: Rotationally invariant body and principal axis

text: Suppose a rigid body is invariant under a rotation around an axis $\mathsf{A}$ by a given angle $0 \leq \alpha_0 < 2\pi$ (and also every multiple of $\alpha_0$). Is it true that in this case the axis $\mathsf{A}$ is a principal axis of the rigid body? If so, how to prove it? Do you have any references for a proof?


title: Why do you only hear high frequencies when a microphone is near its speaker?

text: The phenomenon I'm talking about is _positive feedback_ , as known from control engineering: when the microphone is too close to its speaker, it can "hear itself", so the signal will be infinitely amplified. This can be very irritating e.g. during a concert. But I wonder: why do you only hear a high-pitched tone when this occurs, and not some kind of white noise? Is it because the system acts as some kind of high-pass filter? And if so, why?


title: How much is important the role of Planck length in the strings theory?

text: this is Planck length: $$\ell_p=\sqrt {\frac {G.\hbar}{c^3}} .$$   1. How much is important the role of this length in the strings theory?   2. is this planck's length or newton's length! or maybe both of them!?


title: Full refraction in fibre optics

text: Well in a problem I had to calculate the maximum amount of "reflections" in a glass fibre optic pipe (index of refraction = 1.3, width of 20 micrometer and length of 1 meter). I am a bit blocked on how to calculate this. I considered first the critical angle $sin(\theta_{cr}) = \frac{1}{n}$ Which came out to be 50 degrees. I thought: well at critical angle the whole light beam is reflected. $\theta_i = \theta_o = \theta_{cr}$. And the length of each "reflection" would simply be $\frac{d}{\tan(\theta_{cr})}$. Resulting in $\frac{l}{d}$ reflections in the pipe. However this came a bit over 60 000 reflections. While the answer should be around 42 000. What am I missing here?


title: Does tritium hydride exhibit measurable spontaneous fusion via proton tunneling?

text: In a fascinating 30 June 2013 article in _Nature Chemistry_ , researchers from the University of Leeds found that when molecules of hydroxyl ( **OH** , a fairly stable radical) and methanol ( **CH$_3$OH** ) are cold enough to adhere to each other in deep space instead of bouncing apart, they combine to form methoxy molecules ( **CH$_3$O** ) and water at rates hundreds of times higher than at room temperature. The reason for this unexpected increase in reactivity is both elegant and simple: By being cold enough to cling together loosely for long periods, the integral of very low quantum tunneling rates becomes large enough to enable appreciable rates of combination of the two molecules, despite the reaction having a high classical energy barrier. Now what I find interesting is that there is a simple equivalent of this "let's stay close for a while" situation in nuclear chemistry. Specifically, it is the formation of an ordinary chemical molecule of two nuclei that have a high energy potential for fusion. The chemical bond becomes the analog of the gentle "let's say close" binding of cold hydroxyl and methanol molecules, and the nuclear fusion reaction becomes the analog of the exothermic recombination of the two molecules, albeit with an enormously higher energy barrier. This analogy leads to only two or three interesting candidates: **TH** , tritium hydride; **TD** , tritium deuteride; and **T$_2$** , a molecule of two tritons. **TH** would seem by far the most interesting, since the probabilities for protons to tunnel over atomic distances are orders of magnitude higher than those of deuterons or tritons. The details will be complicated by the larger fusion cross sections of **D-T** and **T-T** in comparison to **H-T** , however. So my question: Has anyone ever specifically looked for this effect -- that is, for higher rates of spontaneous fusion in **TH** that can be attributed to increased tunneling probabilities due to the proton being kept at an atomic distance from the triton for an indefinitely long time? I did a cursory online-only look for relevant articles, without much success. Nonetheless, my best guess is that spontaneous **TH** fusion rates were likely calculated out to the $n^{th}$ degree decades ago, and likely experimentally verified to the same degree. After all, everything in nuclear chemistry is about tunneling probabilities. Nonetheless, the close analogy of long-term atomic-distance binding of nucleons to this more recent finding on increased tunneling between cold-bound molecules is intriguing. I was curious whether anyone out there is familiar with the topic and can talk about it in this forum.


title: Would it be interesting or useful to map objects' current (predicted) locations?

text: Due to light travel time, we observe astronomical objects as they were in the past. If we knew objects' motion (relative to us), it seems like we could extrapolate to their present positions, but I can't remember ever reading about anything like this. Has there been any work like this done--or is there some reason why it would be pointless?


title: Quantum gravity and relevant/irrelevant operators

text: I am familiar with the casual dichotomy in QFT between coupling with positive dimensions in energy implies relevant operator on one side and negative dimension implies irrelevant operator on the other side (forgetting about marginal operators) once you have set $\hbar$ and c equal to 1, and then I thought than in a theory of quantum gravity you would have also set G equals to 1 (or $8\pi G = 1$), so it's no more possible to count in energy dimensions, as there are no more dimensions (scales). Does it only mean that our little trick is now over, or is there any conceptual novelty regarding the RG flow? Thanks


title: Identifying objects, for dummies

text: If I'm a tyro and have a latitude, longitude, time, date, height from the horizon, and compass direction, what means can I use for identifying what I see there? Realize that, as a tyro, I don't have any astronomy books or programs, and don't want to buy any; I might download a free program for my occasional inquiries about the sky, but would prefer a Web site.


title: Generated wavelength of free electron laser

text: Could you please help me understand how one can measure output frequency of free electron laser **(provided that we know size of magnetic domains and electron energy)**? This should be a function of magnetic domains size & electron velocity? But electron velocity should drop due to photon emission - does that means that FEL have a "wideband" spectrum? Why military users focus on Röntgen radiation while UV in air transparancy window might have less air absorption? Bold is still unanswered :-)


title: What makes the difference between ionic and covalent bonds?

text: Backstory: When I learned about chemical reactions, there were two types of molecular bonds: Ionic, where an atom or compound molecule with a low valence number loses those valence electron(s) to one or more other compounds with a high valence number, which it then bonds with; and Covalent, where valence electrons are "shared" between two or more atoms that don't want to actually "give up" their electrons to the other atom. At first, that worked. But then, as I casually poked around Wikipedia and other sites looking to further my knowledge, I found evidence refuting this clear division. For instance, the explanation of pH on Wikipedia states that pure liquid water, a substance which is a textbook example of covalent atomic bonds, isn't actually covalent, but an equal mixture of hydronium (H3O+) and hydroxide (OH-) ions. The explanation is that liquid water, as a polar solvent, actually _dissolves itself_ , and hydrogen ions (naked protons) travel relatively freely among atoms of water in its liquid and even gaseous states, forming hydronium where the hydrogen atom is gained, and hydroxide where it's lost. Another example; benzene. It's composed of 6 carbons and 6 hydrogens. Carbon wants to get four electrons to fill its outer valence shell. The hydrogen fills one, ostensibly by sharing. It gets the other three by sharing electrons with its two neighbors. Now, three electrons don't divide by two; classically, we're taught that the double and single bonds alternate, but really, the behavior of this construct indicates that the last valence electron of each carbon atom is itself shared between its two neighbors, giving each carbon "one and a half" bonds with either of its neighbors. That sounds like "sharing", but not in the way normally described. Now, there are other pretty concrete examples of covalence; diatomic gas molecules like O2 and N2 each have a mostly-full valence shell and just want a couple more valence electrons, and so in their diatomic bonds each atom "pulls" the other's valence electrons into its own valence shell without losing its own. But, "highly covalent" bonds like this seem to be more an exception to the rule than a rule unto itself. So, maybe from a deeper level, I'm asking: where is the "dividing line" between bonds called "ionic" and bonds called "covalent"? Is there a primary property of the bond or the resulting substance that can be given as evidence to say definitively "this is a(n) **___ __ ___ __** bond? Or is there indeed some "grey area" where the behavior of electrons between atoms could be described both ways at the same time?


title: Is this a correct addition to a pipe friction loss calculation?

text: I've been using this online calculator to work out the difference in pressure and flow between two ends of a pipe: Pipe Friction Calculation for Fluid Flow in a Pipe So far most of it makes sense, and I've managed to figure out the worst of it (pipe relative roughness >_<). The main problem is that we don't have the "Average fluid velocity in pipe $\mathrm{V}$", only the Flow at A in $\mathrm{l}/\mathrm{s}$. Before you mock the next part I should explain that I'm a software developer, which apparently still qualifies me more for this sort of work than my water engineer brother-in-law... Anyway, to convert from Flow to Velocity, I made up the following:               L: litres per second at A     D: diameter of pipe (m)     V: fluid velocity (in m/s)      $$V = \frac{ 1000L }{ \pi(D/2)^2 } $$ Is this the correct conversion, or have I missed something?


> **Possible Duplicate:**   >  Age of the Earth and the star that preceded the Sun How were the heavy elements from iron to uranium made? References: http://www.phy.anl.gov/accelerator_rd/index.html Review of proposed Tech: http://www.annualreviews.org/doi/abs/10.1146/annurev.nucl.55.090704.151604


title: Branch cuts in two-point function

text: The propagator of a QFT is known to have a branch cut as a function of the (complex) external momentum. The branch point (as done by, say, Peskin & Schroeder in eqn.7.19 section 7.1) is identified as the root of the argument of the logarithmic piece. Is this not a scheme dependent piece? At least, at the outset it looks so and it is also the same that one gets under dimensional regularization. Is there a general argument to prove that it is scheme independent?


title: What are threshold corrections?

text: _As the title goes, what are **threshold corrections** in quantum field theory?_ In particular, I would be glad if a good reference is provided. Standard QFT books such as Peskin, Weinberg, etc seem to have nothing to say about them.


title: An identity for spinor helicity formalism

text: I have a question about the spinor helicity formalism from arXiv:1308.1697 Denote the massless spin-1/2 fermions as Eqs. (2.10)-(2.11) in that paper $$v_+(p)= \begin{pmatrix} |p]_a \\\ 0 \end{pmatrix} $$ $$v_-(p)= \begin{pmatrix} 0 \\\ |p \rangle^{\dot{a}} \end{pmatrix} $$ $$\bar{u}_-(p)= (0, \langle p |_{\dot{a}})$$ $$\bar{u}_+(p)= ([p |^{a},0)$$ For real momenta, there is an identity in that paper $$ [k| \gamma^{\mu} |p \rangle^*= [p|\gamma^{\mu}|k \rangle \tag{2.33}$$ My question is, how to prove (2.33)? I know $$[p|^a=(|p \rangle^{\cdot{a}})^*, \langle p |_{\dot{a}} = (|p]_a)^* \tag{2.14}$$ for real momenta. By using (2.14) I got $$[k| \gamma^{\mu} |p \rangle = ([p|)^* | \gamma^{\mu} (|k \rangle)^* $$, since $\gamma^{\mu*} \neq \gamma^{\mu}, \mu=2$, I still miss a complex conjugation...


title: Do bubble universes have a centre?

text: This question is motivated by Big Bang Physics/Cosmology. If eternal inflation is a good description of the universe and we live in a bubble universe then presumably our bubble nucleated at a spacetime point. Does this mean it is (approximately) a finite hypersphere and therefore has a centre?


title: Assuming that the Cosmological Principle is correct, does this imply that the universe possess an empirically privileged reference frame?

text: OK...before everyone blasts this with references to the relativistic invariance of the physical laws, time dilation, etc let me add some context. Also, I am an _amateur_ with an interest in physics, so I don't know the details of the physics. However, I have read enough about the cosmological principle and cosmic time to develop this "confusion" . Based on evidence gathered to-date, it appears that the universe is both isotropic and homogenous to a high degree, as evidenced by observations and the usefulness of the Freedman-Lemaitre-Robertson-Walker (FLRW) metric. This principle, when combined with the FLRW metric, allows space-time to be divided into non-intersecting slices and therefore establish a "cosmic time." Now, the existance of such a time in no way suggests that it is the "right" time or a true "now", hence violating relativity -- although some presentist philosophers (e.g., W.L. Craig) have tried to make this argument, but it is not generally accepted in philosophical circles. However, this leaves me confused about an apparent disconnect between relativity and the cosmological principle: Isotropy only holds if we are at rest relative to the cosmic microwave background radiation (universal rest frame), o/w anisotropy is present - yet each reference frame is supposed to allow equally valid observations. How can these two be reconciled when most reference frames would lead us to conclude that the universe is NOT isotropic? It seems that only by appealing to the idea of being at rest relative to "universal rest frame" can we explain away any discrepancies from isotropy as due to "peculiar motion". However, doesn't this give this "universal rest frame" and its associated time an _empirically_ privileged status, even though physical laws work just fine in every reference frame? For example (pardon any abuse of astronomy): if look off in some part of space and see only quasars, and then in another part of space and see only brown dwarfs, but I measure both as apparently the same distance from me, then can I conclude that we are in motion, since otherwise we would have an empirical contradiction (kind of like finding dinosaur and human remains in the same strata)? Any help on where I am going wrong would be helpful. Intuitively, I don't think there should be a way to establish absolute ordering between space-like separated events, but the above reasoning suggests otherwise.


title: What exactly is the meaning of weak formulations and what is its purpose?

text: What is the purpose behind weak formulation of PDEs? I have read in the book by Zienkiewicz and Taylor that a weak formulation is more "permissive" that the original problem in the sense that it allows for discontinuities in coefficients of PDEs where as the original PDE in differential form _must_ have a "truly" smooth solution to be solved analytically. So my question is, firstly what is a weak formulation in the physical sense? and secondly, is it only used to handle PDEs with coefficient discontinuities in the sense that an analytical integration gives a "too smooth" result which isn't relevant in problems of nature?


title: How come that an Infinite universe will collapse under gravity?

text: In this PSE post the issue is about the stability of an infinite universe under Newtonian gravity.   Here I'will drop the Newtonian constraint because we know of the finite speed of interactions and I ask :   How can someone explain the existence of a gravitational collapse ? In the drawing the universe is isodense and infinite and it will extend outside of the limits of the screen ;)   The grayed spherical regions represent the observed universe in relation to the points A,B,C (that can be considered as centers of the universe) and the three points are causally disconected because it is impossible that the interactions had time to propagate from A to B , ... to the entire universe.   You can imagine a similar region around the point X that can see both A and B. Because nowhere exists a gravitational gradient, whatever the density, there is no chance of motion whatever the selected point. IMO. How can someone explain the existence of a gravitational collapse ?   ![enter image description here](http://i.stack.imgur.com/gW9wc.jpg)


title: Uncertainty principle and the energy-momentum 4-vector

text: In each of the uncertainty relations $$\Delta p_x \Delta x \geq \hbar/2$$ $$\Delta p_y \Delta y \geq \hbar/2$$$$\Delta p_z \Delta z \geq \hbar/2$$$$\Delta E \Delta t \geq \hbar/2$$ the second term on the left side is one of the compoments of a position 4-vector, while the first term seems to be the corresponding component of the momentum-energy 4-vector, i.e. the proper time derivative of the position component multiplied by the invariant mass. Does this have any "deeper" (whatever that means) physical significance or something else is going on here? I'm perfectly aware that this might be a silly question, but I'm still learning both quantum and spacetime physics and I never had to deal with the intersection between those two, so I would appreciate a simple clarification!


title: Does the warp drive theorically allows travels between other universes?

text: Does the warp drive allow to travel between universes? Even in a level 2 multiverse?


title: What is the relevance of potential energy as a concept?

text: As far as i understand, for an object to have a fixed quantity of potential energy, it must be in current equilibrium and have no net force applied to it (otherwise the potential would convert to kinetic). For example, i picture a ball on a hill as a classical example. And as far as i know, potential energy also requires some external perturbation to convert it into kinetic energy (i.e. someone pushing the ball). But what can be said about potential energy where two fundamental forces are pushing directly against eachother? i.e. the ball has rolled to the ground, but it still has a gravitational force vector pushing it downwards, however it is balanced by an equal and opposite electromagnetic force vector. The potential is still there because the force vector exists, but it would be almost impossible to do anything useful with it because of the nature of the equilibrium. So here is the conclusion I'm making that i'd like to check the correctness of: Potential energy is just a measure of the force vector applied by a fundamental force on an object, and the amount of "perturbation" that one has to do to turn that energy into kinetic energy is completely arbitrary, thus some potential energy is very functional (requires only a small push to get the ball rolling) whereas other equivalent potential energy would require even more work done on the system to turn the potential into kinetic, thus rendering it "unharnessable". Edit: Ive just realized that since force fields arent homogeneous, a force vector at a given point wouldnt adequately represent potential energy. But aside from that correction, the underlying inquiry i have still remains.


title: Spontaneous symmetry breaking and time-reversal symmetry

text: In most textbooks on field theory you read that "spontaneous symmetry breaking implies degeneracy of the ground state". (Like for example in http://www.itp.phys.ethz.ch/research/qftstrings/archive/13FSQFT2/Chapter07.pdf) I am feeling somewhat uneasy about this. Say you have a system with time- reversal symmetry. Then Kramers Theorem implies that the ground state is at least two-fold degenerate. Now when I break this symmetry spontaneously (or explicitly) I expect that this degeneracy is removed. (Since Kramers Theorem is no longer valid) Where is the logical error that I am making? * * * Edit: Maybe I am also giving a reference to a more advanced application of the question. The Wikipedia Article on Symmetry Protected Topological Order states that "If the boundary is a gapped degenerate state, the degeneracy may be caused by spontaneous symmetry breaking and/or (intrinsic) topological order."


title: Lagrangians not of the form $T-U$

text: My Physics teacher was reluctant to define Lagrangian as _Kinetic Energy minus Potential Energy_ because he said that there were cases where a system's Lagrangian did not take this form. Are you are aware of any such examples? Update: Here I'm of course assuming that $T$ and $U$ stands for the kinetic and the potential energy, respectively. Also:   1. adding a total time derivative term to the Lagrangian, or   2. scaling the Lagrangian with a non-zero multiplicative constant do not change the Euler-Lagrange equations, as Dilaton and dmckee points out in the comments. Needless to say that I'm not interested in such trivial modifications (1&2).


title: Ratio of horizontal to vertical kinetic energy in ocean

text: What is the ratio between horizontal and vertical kinetic energy in the ocean and what is the scale dependencies of this ratio ?


title: $U(1)$ beta function of low energy effective Seiberg-Witten theory

text: My question is about figure3 (page 8) of this paper hep-th/9705131. Start from Seiberg-Witten theory, integrate out the charged high energy modes down to Higgs scale and we get a $U(1)$ gauge theory without matter. Then the author claims that the beta function is zero. Is there any argument or calculation for this? It seems contradict with formula (2.7) Arkani-Hamed and Murayama (hep- th/9707133). $$\frac{1}{g^2_{eff}}= \frac{1}{g^2_{h}}+\frac{1}{4\pi^2}\log{\frac{v^2}{M^2}}+\rm{non-pertubative} $$ ![enter image description here](http://i.stack.imgur.com/XdGOD.jpg)


title: can hydrogen stay frozen in vacuum?

text: I've look into the hydrogen state diagram, and it seems that it can be frozen under pressure. > **Question:** Does this mean that hydrogen cannot be kept frozen in a vacuum > chamber?


title: Explanation of NMOS processes

text: ![Enter image description here](http://i.stack.imgur.com/5rbRT.png) Gate (poly-Si + $SiO_2$) and $p^{-}$ silicon operate as capacitors. But how are voltages and charges applied? In order for inversion to occur, there should be charges formed in poly-Si, right? How does it work? Or if I am mistaken, what is the process? (Both NMOS and PMOS would be appreciated.)


title: Is there an equation for the residual strong nuclear force?

text: First of all, note the qualifier "residual". The present question is not the same as that asked, and answered, in the StackExchange question "Is there an equation for the strong nuclear force?" which was really addressed as though it asked "Is there an equation for the strong interaction between quarks?" If there is no such equation, is there a numeric simulation program?


title: Different directions of frictional force when objects are rolling

text: My textbook has two instances of rolling bodies (smooth rolling). In the first, the body is rolling on the horizontal floor with some acceleration of its centre of mass. In this case, the book says that the friction will act in the direction the ball is accelerating. In the other instance, the ball is rolling down an incline. In this case, the book says that the force of friction acts opposite to $mg\sin\theta$, the component of the gravitational force parallel to the incline. I don't understand the difference. I mean, in both the cases, the ball experiences an acceleration due to a force. Why does the direction of friction change? For me the second case was more intuitive. The friction and gravity both induce torques of the same sign on the body. But I'm lost on the first case. The force that produces the acceleration and the frictional force clearly induce opposite torques which should cancel. Or atleast cancel partially. Also, my intuition says that for the first case, friction should act opposite to the direction of acceleration. I tried to get answers on the web, but I couldn't find anything that was explained in a lucid manner. Some websites mentioned that the direction of friction is affected if the force is applied at the axle. I don't see why that should be.


title: Chirality, helicity and the weak interaction

text: From what I'm understanding about Dirac spinors, using the Weyl basis for the $\gamma$ matrices the first two components behave as a left handed Weyl spinor, while the third and the fourth form a right handed Weyl spinor. By boosting in a direction or in the opposite, I can "asymptotically kill" either the left or right handed part of the (massive) spinor. Since only the left- handed part interacts with the weak force, does that mean that when I see an electron travelling very fast in one direction (same as/opposite to spin) I see/don't see it weakly interacting? This sounds very odd indeed. I have two hypotheses:   1. Massive spinors don't have an intrinsic chirality (since they are not eigenstates of chirality operator), the only information I have is about helicity, and the odd thing I described before is actually observed (really odd to me).   2. Massive particles have an intrinsic chirality, but I don't see how the chirality information gets encoded into the Dirac spinor / how the weak interaction couples to only half of it. To me it seems that only the helicity information is carried by a spinor.


title: DIY Quantum Eraser Experiment by the Scientific American: Is this really quantum?

text: Click here for the publication. Having performed this experiment, I have gotten clean results. Essentially, a double slit is made by putting an photon beam in the way of a wire with orthogonal polarizers on either side. This destroys the expected interference pattern since the polarized filters "measure" the path of the photons. However if one places a 45 degree polarizer that allows the orthogonal light waves to both pass through, the interference pattern restores. According to the article, this is a "quantum eraser" since the wave nature was destroyed with the perpendicular polarizers and restored afterwards with the 45 degree filter. This being said, I also understand that the classical Fresnel-Arago laws state that orthogonal waves do not interfere. Wikipedia also mentions that when [particle detectors are at the slits][], the wave function should collapse. But it also states that this experiment has never been published. Here we have an experiment that places a "detector" at the slits, and as far as Scientific American says, it has collapsed and even restored the wave function. Now, I can only think of 2 conclusions to this: 1) The Fresnel-Arago Laws were a precursor to quantum mechanics and there is no interference because the information has been leaked into the outside environment 2) This is purely a classical experiment and can be explained as such Is this experiment just a demonstration of classical optics or is there actually a quantum nature to this? I also wonder if Fresnel and Arago had an explanation to the nature of orthogonal light waves, or if the quantum mechanical wave collapse due to observation is the only reason. Does anybody have information on this? Much gratitude for your thoughts! This is for a science fair project for my high school, so I would greatly appreciate it since I no longer know whether I should present it as a classical twist to the double slit experiment or a true quantum mechanical phenomenon.


How would a physicist measure temperature of molten metals in 1850-1920s? What equipment would be used?


title: Can the overall sign of the Minkowski metric be changed?

text: If we take the Minkowski metric, $\eta_{\mu\nu}=(1,-1,-1,-1)$, instead of the usual $(-1,1,1,1)$, does this change the form of the Lorentz Transform? I think the standard Lorentz Transform looks like: $$ \begin{matrix} \gamma & -\gamma\beta & 0 & 0\\\ -\gamma\beta & \gamma & 0 & 0\\\ 0 & 0 & 1 & 0\\\ 0 & 0 &0 & 1 \end{matrix} $$


title: Darwin term and Zitterbewegung

text: I've noticed that in the discussion of the fine structure of Hydrogen atom standard QM texts claim that the Darwin term, which corrects energy of $\ell=0$ (or $s$-) states only, is related to the zitterbewegung of the relativistic electron. I'm quite familiar with the zitterbewegung phenomenon, but it is not clear to me how it leads to the correction to the Hamiltonian in the form: $$H_\text{Darwin}=\frac{\hbar^2}{8m^2c^2}\nabla^2 V$$ Furthermore, this article: http://arxiv.org/abs/atom-ph/9607005 claims that the Darwin term has its origin in the spin-orbit coupling rather than zitterbewegung. Who is right? Question:   1. What is the physical origin of the Darwin term?   2. What is the history behind the Darwin term?


title: Working with $\delta$s to use principle of virtual work

text: I'm trying to do the following problem: A lever $ABC$ (see figure) has weights $W_1$ and $W_2$ at distances $a_1$ and $a_2$ from the fixed support $B$. Using the principle of virtual work, prove that a necessary and sufficient condition for equilibrium is $$W_1a_1 = W_2a_2$$ My attempt: Now, the principle of virtual work states that $$\sum_{i=1}^{N}{\bf F}^{(a)}_i\cdot \delta{\bf r}_i = 0$$ I know the forces are equal to the given weights. However, I have a slight problem with the $\delta r_i$. I know it is a variation of the position vector $r$ of the $i$th particle. So, taking the origin as B, we have $\delta r_1=-\delta a_1$ and $\delta r_2=\delta a_2$, and my equation becomes: $${W_1}\cdot(-\delta a_1) + {W_2}\cdot \delta a_2= 0$$ Is my work up to this point correct? And is it correct to simply conclude from here that $W_1a_1=W_2a_2$? I would be more comfortable if I could get rid of the $\delta$s, but given that I'm not comfortable with working with them, I don't quite know how to do this. ![enter image description here](http://i.stack.imgur.com/NSHA4.jpg)


title: Is there any good gauge-fixing prescription for discrete gauge symmetries?

text: Nearly all gauge-fixing prescriptions are based upon setting some function involving the gauge fields to be zero. That function is continuous and varies over the real/complex numbers. Trying the same trick for discrete gauge symmetries break down because we have no continuity, and hence, no implicit function theorem. So are there any good prescriptions out there?


title: The definition of Density of States

text: The density of states (DOS) is generally defined as $D(E)=\frac{d\Omega(E)}{dE}$, where $\Omega(E)$ is the number of states. But why DOS can also be defined using delta function, as $$D(E)~=~\sum\limits_{n} \int \frac{d^3k}{(2\pi)^3}\delta(E-\epsilon_n(\mathbf{k}))?$$


title: Aircraft nose-up glide

text: In the pilot's introductory book "Stick and Rudder" it claims that a nose-up glide is possible. It doesn't state how, why or when. It implies it's possible to do and maintain a constant forward velocity. Is this possible? I really don't see how, unless the aircraft has what I assume would be an extremely unusual design, where the wings would have to have a reverse angle of incidence of the common designs. Glancing through clancy's Aerodynamics, it seems that the force of lift acts upward, and slighly behind the normal of the chord. Given a glide has no thrust, I can't see how the net forces could balance with the drag to maintain forward velocity if the nose is up. Thanks


title: Why doesn't the anthropic principle select for N=2 SUSY compactifications with an exactly zero cosmological constant?

text: The party line of the anthropic camp goes something like this. There are at least $10^{500}$ flux compactifications breaking SUSY out there with all sorts of values for the cosmological constant. Life takes a lot of time to evolve, and this is incompatible with a universe which dilutes away into de Sitter space too soon. The cosmological constant has to be fine-tuned to the order of $10^{123}$. Without SUSY, the zero point energy contribution from bosons and fermions would not cancel naturally. However, superstring theory also admits N=2 SUGRA compactifications which have to have an exactly zero cosmological constant. Surely some of them can support life? I know there are a lot more flux compactifications out there compared to hyperKahler compactifications, but does the ratio exceed $10^{123}$? What probability measure should we use over compactifications anyway? Trying to compute from eternal inflation leads to the measure problem.


title: Special Relativistic Time Dilation -- A computer in a very fast centrifuge

text: Ok, I've stumbled onto what I think is a bit of a paradox. First off, say you had some computer in a very fast(near light speed) centrifuge. You provide power to this computer via a metal plate on the "wall" of the centrifuge's container, so it works similar to how subways and streetcars are powered. If the computer normally would consume 200 watts, how much power would it consume at say 1/2 of light speed? Would it consume 400 watts from our still viewpoint? Also, what if you were to be capable of communicating with this computer? Would the centrifuge-computer receive AND transmit messages faster from our still viewpoint? I'm a bit lost in even thinking about it.


title: In What Frame of Reference does the Special Theory of Relativity Operate?

text: > **Possible Duplicate:**   >  Time Dilation - How does it know which Frame of Reference to age slower? This has bugged me for years. According to the theory of relativity, the faster an object moves, the slower time goes for that object, relative to a stationary observer. However, it is my understanding that motion is necessarily relative to a frame of reference. If you have two objects in space moving away from each other at a combined velocity of 20mph, the rate at which each is moving is totally conditional on what frame of reference you specify.   1. If object A is your frame of reference, than object A is moving at 0mph, and object B is moving at 20mph.   2. If object B is your frame of reference, the opposite is true.   3. If something else is your frame of reference, then the two objects will have some other two velocities, which, when summed, equal 20mph. If this is true, how does physics know how much to slow down time for each object? It's as if they don't actually have set velocities.


title: Standing Wave: How to show that $\frac{\mathrm df}{f}=0.5\frac{\mathrm dF}{F}$ where $f$ is frequency and $F$ is tension of the string?

text: How to do this? Show that if the tension $F$ in a string is changed by a small amount $\mathrm dF$, the fractional change in frequency of a standing wave, $\frac{\mathrm df}{f}$ is given by: $$\frac{\mathrm df}{f}~=~0.5\frac{\mathrm dF}{F}.$$


title: How much (little) wind force does it take to move a toddler in a stroller?

text: This question is raised because of the recent incident in a London subway station involving a stroller falling onto the tracks. The stroller was probably 10 feet (or 3 m) from the edge of the platform but was moved by the wind as the train departed and went into the tunnel. Assume the weight will likely be 10 kg in total. Also, \- what if the wheels of the strollers are locked? \- What if they are unlocked?


title: Angular Frequency about stable equilibrium

text: So I have the potential energy defined as: $$U(x)=U_0[2(\frac xa)^2-(\frac xa)^4]$$ And I want to find the angular frequency of oscillations about the point of stable equilibrium. I found the point of stable equilibrium to be 0, but now I do not know how to find the angular frequency. Normally I would look to use $$E=K+U=\frac 12m \dot x^2+U(x)$$ But I do not know where an $\omega$ would end up popping out. I a not looking for a solution, just a hint. EDIT:: My attempt: Using conservation of energy, I get that $$\frac 12 mv^2=U(a)=U_0$$ And then $$\omega=kv=\frac {2\pi}{\lambda}\sqrt{\frac{2U_0}{m}}$$ Where $\lambda$=2a


title: Asymmetric Schwarzschild solution - unequal mass on each side

text: Look at any Kruskal–Szekeres coordinate plot of the Schwarzschild solution. It shows the same mass everywhere. Yet the two sides cannot talk to each other, in that no information, particles, etc can cross the wormhole throat. So how do the sides 'know' to be the same mass? So is there a way to draw a Kruskal–Szekeres plot with the masses unequal on each side? In other words, would the geometry of space play nice and smooth at the interface between regions II and III, where different mass solutions are right next to each other? Another way of putting this is that if you overlap two K-S diagrams with different mass, M1 in region I && II, and M2 in region III and IV, and then do an embedding diagram, will you see something different than the single mass version. Another way of putting it. The Schwarzschild solution is static, and unique. So can you sew two of them with dissimilar masses together coherently? If not, then it would seem that another - non static - solution is in order, which would be surprising, since there is only one parameter (M) to be non static. Look at say http://www.csun.edu/~vcmth00m/embedding.pdf or similar for diagrams.


title: What kind of manifold can be the phase space of a Hamiltonian system?

text: Of course it should have dimension $2n$. But any more conditions? For example, can a genus-2 surface be the phase space of a Hamiltonian system?


title: What is the pressure of a gas required to ionize the gas using an electron gun?

text: How dense does a gas (Argon in particular ) have to be to in order to ionize it using electron bombardment and weak magnetic fields. Is there a correlation with the density of a gas and the easiness to ionize the gas ?


title: Is quantum tunneling related to imaginary time?

text: I was studying for my exam and looking at the chapter which talks about Potential-energy graphs. Let's take this as an example:   ![Potential-Energy Graph](http://i.stack.imgur.com/F0iVj.jpg) My book states that: "If the object is in $B$ and has a total energy of $0$ then it can only vibrate between the points $A$ and $C$." Which makes sense because if it went beyond e.g. $C$, that would mean $U > E$, which is 'impossible' because then $K<0$. But having read about imaginary time and quantum tunneling (I don't really understand the concepts though.) I had the following thought: If $K<0$ that would mean that I have an imaginary value for my speed $v$ and since imaginary time exists that could mean $t$ has an imaginary component. It made sense in my head because I somewhat understand that quantum tunneling means a particle can get to certain positions which their initial position and energy wouldn't allow in classical mechanics. Are the two at all related or is this too farfetched and totally unrelated?   I checked Wikipedia and didn't find much.


title: Deriving Feynman rules from a Lagrangian for vertex factors for "more complicated" interactions

text: I am trying to derive Feynman rules from a given Lagrangian and I got stuck on some vertex factors. What for example is the vertex factor that corresponds to the four-scalar interaction that is decribed by the following Lagrangian? \begin{equation} L = -\frac{1}{4} g_3^2 \phi^\dagger \lambda^a \phi \chi^\dagger \lambda^a \chi + \frac{2}{9} g_1^2 \phi^\dagger \phi \chi^\dagger \chi \,, \end{equation} where $\phi,\chi$ are complex scalar (color triplet) fields, $\lambda^a$ are the Gell-Mann matrices, and $g_1,g_3$ are the coupling constants corresponding to $\text{U}(1)$ and $\text{SU}(3)$ respectively. If we would have only had the second term here, say, then the vertex factor would simply be found by "dropping" the fields and multiplying by $i$. But now there are two terms contributing, and in the first term the Gell-Mann matrices even mix the color components of the scalar triplets. So how do I proceed in this case? And could anyone give me some general strategies on how to derive vertex factors for "complicated" interactions? For example, I also find it tricky to get the sign right if there is a derivative in an interaction. (If you are interested in the context of this Lagrangian, for $\phi = \tilde{u}_R$ and $\chi = \tilde{d}_R$ this Lagrangian describes the interaction between two up squarks and two down squarks in a supersymmetric theory.)


title: How to model energy loss in a rotating body?

text: I recently asked a question about modeling instability in a rotating rigid body. I now realize that I was mentally confounding two different effects:   1. The "Dzhanibekov effect" in which a rigid object with three different moments of inertia appears to tumble when spun around the intermediate axis. It ends up oscillating in a rather complex-looking pattern.   2. The tendency of an object (e.g. a fluid-filled cylinder) to change its spin axis to that with the greatest moment of inertia. I've successfully reproduced effect 1, which in the end is a relatively simple (if somewhat surprising) result of conservation of angular momentum. Effect 2, however, is _not_ something that occurs with ideal rigid bodies. It happens only when there is some mechanism for energy loss -- say, whipping antennas (as in the famous Explorer 1 satellite), or movement of an internal fluid (as in this video). I've found explanations saying that in these cases, rotational energy (aka angular kinetic energy) is lost, though angular momentum remains (in some way) unchanged. I'd like to model this effect. I imagine it is a matter of transferring some momentum from one axis to another, but in what way?


title: Angular momentum in an accretion disk

text: I need to plot the time evolution of the total angular momentum in an accretion disk. This confuses me because I thought this should be constant, since angular momentum has to be conserved? I'm given the angular velocity $\Omega=(GM/R^3)^\frac{1}{2}$ where $M$ is the mass of the central object, and that the disk is made up of annuli of matter lying between $R$ and $R+\Delta R$ with mass $2\pi R\Delta R\Sigma$, where $\Sigma(R,t)$ is the surface density at time $t$ (I calculated the surface density numerically at different times in the previous question, so I assume this has to be used in my answer). So a) Why does total angular momentum change? and b) How do I know what function of $R$, $\Omega$, and $\Sigma$ represents total angular momentum?


title: Radiofrequency attenuation in cold plasma

text: The missile plume can be considered a cold plasma. If a radio signal passes across the plume it's attenuated. Obviously the attenuation depends on the frequency of the signal. Where can I find some information about the behaviour of the RF across the cold plasma? Thanks


title: How was the definition and the formula of work derived? Is it the best possible?

text: Work done is defined as the dot product of force and displacement. However, intuitively, should it not be the product of force and the time for which the body is acted upon by the force (force * time) because while time is independent of force applied, displacement is not. Were these formulae (for work and energy) actually derived based on some physical understanding or are they just constructs to understand forces better?


title: Forced vibration

text: Let's consider a spring which is subjected to forced vibrations: $$ F = F_0 \cos(\omega t) $$ Is the resonance frequancy $\omega_0$ of the spring dependent on the amplitude $F_0$? I ask this because I am currently conducting tests with a plate which is forced to vibrate in the Z-direction orthogonally to its plan, thanks to a shaker, and it turns out that the resonance frequency of the plate is different for different values of the shaker amplitude (a higher amplitude gives a higher resonance frequency) Thank you.


title: Local number operators in quantum field theory

text: Redhead claims in his paper "More ado about nothing" (http://link.springer.com/article/10.1007%2FBF02054660) that number operators associated with different space points (at fixed time) fail to commute, and hence are not physically meaningful. However, Halvorson, in his paper "Reeh-Schlieder defeats Newton-Wigner" (http://arxiv.org/abs/quant-ph/0007060), section 3.1, claims that operators $N(x)=a^\dagger(x)a(x)$ are not even mathematically well-defined. However I can't understand in what sense his argument using phase invariance proves that such operators are not well defined: we are simply taking the product of two unbounded operators. This product might indeed not have a clear physical sense (more precisely no "nice" localisation properties), but this was more or less Redhead's claim. So basically I'm trying to understand if $N(x)$ is not associated to any local algebra and hence is not physically meaningful or really mathematical ill- defined, and if so what would be a clear argument to prove it.


title: What does the ordering of creation/annihilation operators mean?

text: When a system is expressed in terms of creation and annihilation operators for bosonic/fermionic modes, what exactly is the physical meaning of the order in which the operators act? For example, for a fermionic system with states $i$ and $j$, $c_i c_j^\dagger$ is different from $c_j^\dagger c_i$ by a sign change, due to anticommutativity. I understand the mathematics of this, but what does it mean intuitively? The former would be described as destroying a particle in state $j$ "before" creating one in state $i$, but what does "before" actually mean in this context, since there's no notion of time? As another (bosonic) example, $a_i^\dagger a_i$ is clearly different from $a_ia_i^\dagger$, since acting the former on a vacuum state $|0\rangle$ gives zero while the for the latter, $|0\rangle$ is an eigenstate, but again, what is the physical interpretation? My normal interpretation of commutativity as a statement regarding the effect of a measurement on a state fails here since creation/annihilation are obviously not observables. I hope the question makes sense and isn't too abstract!


title: Solution Maxwell's equations cylinder

text: I face some trouble solving Maxwell's equations inside a cylinder with perfect conductor boundaries (in 3D) ? We work with cylindrical coordinates $(r, \phi, z)$ and we make the assumption that fields have a sinusoidal "$e^{i\omega t}$" time dependence. Note that we have a $\phi$ symmetry. First, and in any coordinates system, by taking the rotational and injecting one equation in the other we reduce Maxwell's equations to the following, $$ \nabla\times\nabla\times E = -\partial_t^2 E = \omega^2 E $$ In vacuum, from the $curl curl$ identity, it leads, $$ \nabla\times\nabla\times E = \nabla(\nabla . E) - \nabla^2 E = - \nabla^2 E $$ Where $- \nabla^2 E$ is the laplacian operator applied to each coordinate. Now, in cylindrical coordinates, we can only compute the $z-$coordinate since, in this case we get the wave equation, $$ \nabla^2 E_z = \omega^2 E_z $$ For the other coordinates, the change of coordinates introduce other terms such that (for the $\phi-$ coordin. ate)$\frac{E_r}{r^2} - \frac{2}{r^2}\frac{\partial E_\phi}{\partial\phi}$. Then, a fastidious step consists in performing a separation of variable which leads us quite easily to the solution for every separated variable and also to the Bessel differential equation which brings its solution, the Bessel function. Together with boundary conditions we can get the solution according to $z$ but what about the other coordinates ?


title: Doubt in law of mutual interaction

text: Book: Classical mechanics (textbook) by Douglas Gregory (cambridge publications) Law of mutual interaction states that when two particle (let it be P1 and P2) interacts, the particle (P1) induces an instantaneous acceleration (a21) on particle P2 and the particle P2 induces an instantaneous acceleration (a12) on particle (P1). If the (inertial)masses of the particles are same, then the magnitude of acceleration be the same, and the ratios of acceleration will be constant ( for this case it is 1)(consistency relation) That is what Newton's third law says. > My question is, for different (inertial)masses the ratio will be constant ( > but not unity) ( it does not satisfy consistency relation) Am i right? > > If yes My question is consistency relation is important in classical > mechanics?


title: How can there be really any instantaneous velocity?

text: I have read about Zeno's arrow paradox that tells us there is no motion of the arrow at a particular instant of its flight. It can be inferred that there can be no velocity at any instant. Moreover we cannot calculate velocity at any instant in the real world (of course it can be done by using calculus) but how can this be possible? What is the intuition behind this concept?


title: What happened to high temperatures in the universe?

text: We know that after the big bang the temperature was about 10^32 K.   But now the average temperature of the universe is about 4 K. What happened to the temperature at that time? Where did it go?


title: What restrictions on time boundary conditions does it have to use Fourier transform to solve wave equation?

text: The wave equation can be solved using Fourier transform, by assuming a solution of the form of $$\mathbf{E}(x,y,z,t)~=~\mathbf{E}(x,y,z)e^{j\omega t}$$ and then reducing the equation to the Helmholtz equation.   * What are the presumed _restrictions_ on the solution, when solving the equation this way? (e.g., on _time_ boundary condition) I mean can this method give the most general solution (given some boundary conditions)? What _features_ does the solution obtained this way have?   * Does it have any difference with solutions obtained using Laplace transform? (The very same above questions, for Laplace transform.)


title: Find contact force given sphere's radii and distance

text: I simulate scenario of two contacting elastic spheres, where their radii and the distance of their centers is known, and repulsive force should be computed. Material parameters are also given. Most contact mechanics references (also the Wikipedia page on Contact mechanics give equations for determining $a$ (contact area radius) when the force $F$ is already known. Simpler contact models such as Hertz are able to express force as function of overlap of undeformed spheres ($F=\frac{4}{3}E^*\sqrt{R^*}\delta^{\frac{3}{2}}$), but the more complicated ones always only give an equation for $a$ supposing $F$ is already known; it does not seem that $F$ can be analytically expressed from those equations - e.g. in the Carpick-Ogletree-Salmeron model with $a=a_0(\beta) \left(\cfrac{\beta + \sqrt{1 - F/F_c(\beta)}}{1 + \beta}\right)^{2/3}$. So, in the case of more complicated models such as the COS one:   1. do I need to find $F$ iteratively so that the equation for $a$ is satisfied, or am I overlooking something obvious?   2. What is the relationship between $a$ and the overlap $\delta$? I will be also grateful for any good reference.


title: Hyperbolic geometry in SR

text: I'm having trouble understanding just what the hyperbolas on a space-time diagram actually signify. From what I understand, these hyperbolas trace out the equation $S^2=x^2-(ct)^2$, which is the equation for a hyperbola. Fine so far. Furthermore, the plot is supposed to represent the value of $S$, which is the invariant interval. When I think of a spacetime event, though, I think of a line that begins at the origin and that moves out the upper right of the diagram, in between the $ct$-axis and the 45% Null line of the speed of light. I can see how different frames of reference could represent different coordinate shifts of the Spacetime diagram which would yield different values of $x$ and $ct$, keeping the value of $S$ invariant. What I am struggling with is understanding what the hyperbolic shape of the plot of $S$ on the spacetime means. What is it saying? How do I read it? Assuming we are only dealing with time-like events, what do different values of $S$ signify? Is there only that single hyperbola that the value of $S$ falls along for a given spacetime event, or is that curve part of a larger family of curves?


title: What would be the effect of an excess of up quarks on stellar formation?

text: Suppose you had 80% up quarks, and only 20% down quarks. How would this affect stellar formation?


title: How can be proved that one-way speed of light is equal to two-way speed?

text: Using Special relativity theory, of course. Can Lorentz transformations to "tell" something about it? "Wikipedia's" article: http://en.wikipedia.org/wiki/One-way_speed_of_light .


In Vladimir I. Arnold's _Lectures on Partial Differential Equations_ , Chapter 3 _Huygens' Principle in the Theory of Wave Propagation_ , which is devoted to the proof of Huygens principle (original one by Huygens, maybe not by Fresnel), it's said that > the contact elements of space-time that are tangent to the large front > belong to the retardation hypersurface. By _contact elements of space-time tangent to the large front_ , he means roughly the tangent hyperplanes of the large front, if the space-time is just simply $\mathbb R^3\times\mathbb R$ (It's generally a manifold $M^{n+1}=B^n\times\mathbb R$, where $B^n$ is the manifold of physical space). By _the large front_ , he means the geometric object in the space-time that describes the poropagation of fronts of perturbations by means of a single hypersurface in space-time with a more general retardation hypersurface in the manifold of contact elements of space-time, or simply just the graph of all fronts in the space-time. By _retardation hypersurface_ , maybe he means _Fresnel hypersurface_ , i.e. a hyperplane in the manifold of co-oriented contact elements of space-time defined by the field of cones of possible velocities. He says that > In geometric optics, _a cone of possible velocities of motion_ is prescribed > at each point of the space-time manifold $M$. For example, in the flat space of vacuum, the light cone is prescribed at each point of the space time. And the points of _Fresnel hypersurface_ are just the tangent planes to the cones of possible velocities of motion in the manifold of contact elements, i.e, the bundle of $PT^*M\to M$ or co-oriented $ST^*M\to M$. In order to understand these concepts, a solid physical maturity seems needed, which I haven't, so I want to fix the gaps in my knowledge of physics. First, > What's the local law of propagation of disturbances in the geometric optics? Second, > What's the physical meaning of the _Fresnel hypersurface_? Is it well-known > or related to other well-known terms? I need some clues. Thanks!


title: What is crystal field anisotropy or effect ? It forces the magnetic moment to point in particular local direction..

text: Can you give a basic explanation of what is crystal field anisotropy ? What is the reason to arise ? In spin ice it forces the dipoles to point in the local 111 direction. For partially filled rare earth atoms hund rule requires S and L max. This leaves (2s +1)(2l+1) degeneracy which is partially lifted from the LS coupling. When inserted in anisotropic field inside crystal the expectation value , or mean value of L is 0 = 0 and the L is quenched leaving only S so is forced to point in some local direction, but this is if the field removes the deneracy ? Is it this or it is much more complicated ?


title: Lightning and nuclear fusion

text: I'm going to be brief, I just saw a Discovery Channel show that showed a lot of interesting phenomena around lightning (like elves, how cool is that(!)), and got me wondering. 1) Thinking of lightning as a purely mechanical phenomenon, I would think the elves are the "other side of the momentum balance". What I mean is this: somewhere in some cloud formation, an event happens that triggers a lightning flash. This means a ton of electrons (and other, associated particles) that start a _very_ fast journey to the surface of the Earth. Their momentum must be balanced by particles going in the opposite direction, hence elves. Am I right? 2) Taking this further, is it possible that the "trigger" for the lightning flash could be atomic/molecular fusion forced by extremely high electrical fields, that may only exist for a nanosecond, causing a ton of energy to be transferred to the electrons around the atoms/molecules, and we have lightning. I would then think of the energy necessary to force the fusion to occur as a quantum fluctuation as in $$\Delta E \Delta t \leq \hbar$$ 3) Is every arc caused by a particle collision, somewhat like in a particle accelerator? It seems logical to me: there are more than enough particles in the air to collide with, and all the light could well be some form of Brehm or Cherenkov radiation. The question I seem to be asking is where to find good scientific theory/information about lightning. Some say we know a lot about it, but I haven't found any good papers explaining it. I have a pretty good background in physics (1st year Master student) and am not afraid of serious literature. Thanks!


title: How do you do an integral involving the derivative of a delta function?

text: I got an integral in solving Schrodinger equation with delta function potential. It looks like $$\int \frac{y(x)}{x} \frac{\mathrm{d}\delta(x-x_0)}{\mathrm{d}x}$$ I'm trying to solve this by splitting it into two integrals $$\int_{-\infty}^{x_0 - \epsilon} \frac{y(x)}{x} \frac{\mathrm{d}\delta(x-x_0)}{\mathrm{d}x} + \int_{x_0 + \epsilon}^{\infty} \frac{y(x)}{x} \frac{\mathrm{d}\delta(x-x_0)}{\mathrm{d}x}$$ and then do the limit $\epsilon\to 0$. Could you tell me how to solve this integral please? I used Mathematica, it gave out a weird result.


title: Proof of Yang's theorem

text: Yang's theorem states that a massive spin-1 particle cannot decay into a pair of identical massless spin-1 particles. The proof starts by going to the rest frame of the decaying particle, and relies on process of elimination of possible amplitude structures. Let $\vec\epsilon_V$ be the spin vector of the decaying particle in its rest frame, and let $\vec\epsilon_1$ and $\vec\epsilon_2$ be the polarization 3-vector of the massless particles with 3-momenta $\vec{k}$ and $-\vec{k}$ respectively. In the literature, I've seen arguments saying that $\mathcal{M_1}\sim(\vec\epsilon_1\times\vec\epsilon_2).\vec\epsilon_V$, and $\mathcal{M_2}\sim(\vec\epsilon_1.\vec\epsilon_2)(\vec\epsilon_V.\vec{k})$ don't work because they don't respect Bose symmetry of the final state spin-1 particles. But, why is $\mathcal{M_3}\sim(\vec\epsilon_V\times\vec\epsilon_1).\epsilon_2+(\vec\epsilon_V\times\vec\epsilon_2).\epsilon_1$ excluded? Sure, it's parity violating (if parent particle is parity even), but that's not usually a problem Thanks


title: Center of gravity of vehicle and vehicle performance

text: I am interested to know how does CG of vehicle plays role in the Fuel economy and the vehicle performance. Does CG of vehicle has anything to do while accelerating of your vehicle. I am a student, henceforth I would feel glad to know about these issues. Can any one give me the link that containing these topics.


title: Poynting Vector for a Laser

text: How do you calculate the Poynting vector for a laser given it's power? I know for a sphere you can just take the power, and divide it by 4$\pi R^2$, but I don't know what I would do for a laser. Would I have to take into account the aperture size?


title: How can I lower the temperature in my fridge compartment to become a high temp freezer?

text: I have a fridge compartment, the lowest temperature can be set to to about 0c. Is it possible I could bring it down to -2c, -3, -4 c etc by simply adding water bottles which were previously frozen in a freezer compartment. Would the temp be similar throughout the entire fridge or would it fluctuate significantly?


title: Vector Components and Resultant

text: My sister got this homework that she don't understand, me neither. Can anyone help out? Here's the problem: Give the x-component and y component of the following vectors: a. A = 7.0 cm, E b. B = 5.7 cm, S c. C = 5.5 cm, 30 degrees E of N d. D = 5.5 cm, 60 degrees S of E Get the resultant of the four vectors, R = A + B + C + D. Thanks in advance :)


title: Can a single classical particle have any entropy?

text: recently I have had some exchanges with @Marek regarding entropy of a single classical particle. I always believed that to define entropy one must have some distribution. In Quantum theory, a single particle can have entropy and I can easily understand that. But I never knew that entropy of a single rigid classical particle is a well defined concept as Marek claimed. I still fail to understand that. One can say that in the classical limit, the entropy of a particle in QT can be defined and that corresponds the entropy of a single classical particle. But I have difficulty accepting that that gives entropy of a single Newtonian particle. In my understanding, If a system has entropy then it also should have some temperature. I don't understand how one would assign any temperature to a single classical particle. I came across a paper where there is a notion of "microscopic entropy". By no means, in my limited understanding, it corresponded to the normal concept of entropy. I am curious to know, what is the right answer. So, my question is, is it possible to define entropy of a single classical particle?