title: Possible intergalactic celestial objects

text: We know there are rouge stars floating in intergalactic space, thought to be caused by galactic collision. What other other classes of celestial object could be found floating around in intergalactic space? Within galaxies there are the following types of object:   * Nebulae    * Stars   * Brown Dwarfs   * White Dwarfs   * Black Holes   * Neutron Stars   * Comets   * Asteroids   * Interstellar rogue planets   * Planetary Systems   * Star Systems   * Star Clusters   * Debris Disks Which of these can possibly exist in intergalactic space? Asteroids and comets are probably quite common, but is it known for entire planetary systems to exist in intergalactic space? Even star clusters? Out of all the possible types, could I also have a link / designation to each object thats been observed, floating in intergalactic space. For example, here's an intergalactic star HE_0437-5439


title: For a planet which has a temperature gradient, hot in the center and cooler on the surface, why do we get absorption lines?

text: For a planet which has a temperature gradient, hot in the center and cooler on the surface, why do we see absorption lines? Similarly, why do we see emission lines if the planet is hot on the surface and gets cooler as you move to the center? Note, for this question I am only thinking of the planet as a black body, not as something, for example, transiting a star and observing the spectra (transit spectra) of the light that shines through the planet's atmosphere. The source of the spectra is the planet itself.


title: Seeking Recommendation for Celestron Firstscope Eyepiece Upgrade

text: I have a Celestron Firstscope telescope and like it overall for my location and the amount of observing I do. I am disappointed in my view of the planets with the scope. What would be a good eye piece to purchase for this telescope to improve planetary observing. In keeping with my limited budget, it should not cost more than the telescope.


title: How is the Hanbury-Brown and Twiss effect used to measure the size of stars?

text: I understand what an Hanbury Brown and Twiss (HBT) interferometer _does_ , but how can this be used to measure the apparent angular diameter of some object? What is the mathematical explaination?


title: Hartmann mask based collimation

text: Is there a detailed description for a Hartmann mask based collimation process? I've been told by a friend that is possible to collimate an SCT by placing a three round holes Hartmann mask at the secondary and then covering one hole at a time. Unfortunately the guy did not clearly describe the full procedure and what I should try to obtain looking in the eyepiece (or looking at an attached webcam video).


title: Birds sitting on electric wires: potential difference between the legs

text: We have seen birds sitting on uninsulated electric wires of high voltage transmission lines overhead without getting harmed, because sitting on only one wire doesn't complete any circuit. But what about the potential difference between their legs? Is this not a small complete circuit? Because the wire has a potential gradient, there should be a potential difference between the bird's feet. Is this potential difference so very small that we can say the bird is sitting at a single point on the wire? If a bird of a sufficiently large size, with a wide gap between its feet, sits on a single wire, shouldn't the bird receive a shock if the potential difference is sufficient?


title: rate of flow out of a reservoir

text: A reservoir contains a fluid, and there is an aperture at the bottom through which the fluid flows out as shown in the illustration: ![enter image description here](http://i.stack.imgur.com/Ne3dy.gif) I understand that the rate of flow is based on at least these parameters: the amount of fluid in the reservoir due to the weight of the fluid, the size of the apperture, and the density of the fluid. But how exactly do these relate to the flow rate? That is: $$Q=f(V,A,\rho)$$ What is $f$?


title: Problem involving 1st law of thermo and ideal gas law

text: Problem: > $1.0 \text{ kg}$ of air at pressure $10^6 \text{ Pa}$ and temperature $398 > \text{ K}$ expands to a five times greater volume. The expansion occurs such > that in every instance the added heat is a quarter of the work done by the > gas. Calculate the final pressure. $1 \text{ kmol}$ has a mass of $29 \text{ > kg}$ and $C_V = \frac 52 R$. I solved the problem but I have some questions regarding the approaches I made. I interpreted heat being a quarter of the work done every instance as $\text{d}Q = \frac 14 \text{d}W$. By the 1st law of thermodynamics ($Q = W + \Delta U$), this yields $-\frac 34 \text{d}W = \text{d}U$. But for molar heat capacity we know that $\text{d}U = nC_V\text{d}T$, hence $-\frac 34 \text{d}W = nC_V\text{d}T$. By definition, $\text{d}W = p\text{d}V$ and therefore $-\frac 34 p\text{d}W = nC_V\text{d}T$. We treat the gas as ideal and thus we can substitute $\frac {nRT}V$ for $p$, arriving at $-\frac 34 \frac{nRT}V \text{d}V = nC_V\text{d}T$. Separation of variables yield $-\frac 34 \frac{nR}V \text{d}V = \frac {nC_V}T \text{d}T$. We take the definite integrales $ \displaystyle \int_{V_1}^{5V_1} \\!\\!\\!\\!\\! -\frac 34 \frac{nR}V \text{d}V = \int_{398}^{T_2} \frac {nC_V}T \text{d}T$ and can then finally solve for $T_2$ (it turns out to be approximately $245 \text{ K}$). At last, we can determine the final pressure from $\displaystyle \frac {p_1V_1}{T_1} = \frac {p_2V_2}{T_2}$. * * * **(1)** My main question is whether you guys know of a different way to solve this problem not involving having to solve differential equations? **(2)** I've seen the 1st law written as $\text{d}U = \text{d}Q + \text{d}W$ but if we use that expression we definitely won't get the right answer; what's up with the inconsistency? **(3)** I saw no other option than having to use the molar heat capacity equation $\text{d}U = nC_V\text{d}T$ but doesn't this assume constant volume? In our problem the volume is definitely changing, so is it not contradictory to use that very equation?


title: Does the expansion of the universe soon after the Big Bang affect the amount of time that light takes to reach us?

text: If faster than light travel is impossible, how is it that light emitted from matter so close together in the time soon after the Big Bang is only now just reaching us? I would assume that there would be a "limit" to how far back we can see, but exactly how long after the Big Bang are we able to observe? I'm sure there's an easy explanation, but it has been bugging me for a while... I am aware of how this works, but I am curious as to how long after the Big Bang are we able to see in images such as the Hubble Deep Field? If light travels at a constant speed no matter the conditions the observer is experiencing, then why would there be a _longer_ stretch of time for light to travel through as the universe expanded? Please let me know if I should elaborate more on my question because I do feel like my writing is a little bit hard to understand...


title: Doubt in problem regarding resistance a of cylinder

text: I've been working on some exercises and I'm in doubt if my procedure with this one is correct. We have a hollow cylinder with internal radius $r_a$, external radius $r_b$, resistivity $\rho$ and length $L$. A difference of potential $V$ is then applied on the extremes paralel to the axis of the cylinder. First I'm asked to find the resitance. Well, for this one I've just used that $R=\rho L/A$ and calculated $A=\pi(r_b^2-r_a^2)$. Second, I'm asked to find the current density when $V$ is applied. Well, what I did was suppose that the cylinder is an ohmic resistor so that $V=Ri$ holds, and then I've got $i=V/R$. Then the current density should be $J=i/A$ and so we have $J=V/RA$ which gives $J=V/\rho L$. Is this correct? Can I assume that the cylinder is ohmic? Third I'm asked to find the electric field. Now I've used that $\rho = E/J$ so that we must have $E = \rho J$ so that $E = V/L$, this says tha that the field is uniform, but I'm a little unsure about it. Finally the fourth one asks to find the resitance again if now the current flows radially from inside to ouside. In this final case I think that it'll change just the cross sectional area, but I didn't find some easy way to write this down. Can someone give a little help with this?


title: Unitary Confirmation

text: I am asked to show that an new defined operator: $$U_{\beta} = \exp(\displaystyle\frac{i\beta L_z}{\hbar})$$ is unitary, where $$L_z = -i\hbar\,\,(x\displaystyle\frac{\partial}{\partial y} - y \frac{\partial}{\partial x }).$$ I tried the following: $$ U_{\beta}^{\dagger} U_{\beta} = \exp \left( \frac{i\beta(-L_z^{\dagger}+L_z)}{\hbar} \right)$$ So I couldn't make the inside of exponential zero.


title: How to imagine the first few moments of an LR circuit?

text: For example, there's a very simple circuit which only contains on resistor. So according to Ohm's law, we have: $\mathrm{emf} = IR$ As we know when time $t = 0$, the current must be $I = 0$. However, how do I describe how the current really behaves just after I switched on the circuit? You may ask why I care about that. Since I'm learning self-inductance. The most torturing part of it is understanding the "back Emf" induced by changing magnetic field. Every textbooks in which I've looked up this part dismiss the detail of how the "back Emf" really impact on the varying current, instead, they just say "back Emf" pulled the current and "slowed" it down, which is quite vague and obscure. And I devised a situation where this vaguely described intuition really burns out my head: In LR circuits, we have the following differential equation: $$\mathrm{emf}-L \frac{dI}{dt}-IR = 0$$ Let's take $ t = 0$ to see what is going on. At the time $t = 0$, obviously we have $I = 0$, which indicates "no current" at all. So $IR$ must also be zero, we therefore have: $$L \frac{dI}{dt} =\mathrm{emf}$$ which means the inductor has produced a "back emf", that is to say, there exists a magnetic field in the inductor. But how? Since there's no current at all. Furthermore, can anyone help me understand the very first moments when an LR circuit is switched on? Thanks in advance.


title: Does the Casimir effect give the correct value for Dark Energy?

text: My understanding is that the Casimir Effect is caused by vacuum energy. Quantum mechanics (QED) predicts vacuum energy, but gets the value grossly wrong, by a factor of $10^{120}$. On the other hand, from what I have read, Dark Energy is understood to be caused by vacuum energy. Has anyone checked whether the measured value for the Casimir Effect fits the value required for Dark Energy to make up 70% of the energy in the universe? If it did, that would be an excellent validation of Dark Energy.


title: Why do we order the variables in certain physics questions the way we do?

text: I'm writing a script involving physics equations, and someone complained that my script outputs $F = m a$ as $F = a m$, as well as outputting $E_p = m g h$ as $E_p = g h m$; another example would be $E = m c^2$ vs $E = c^2 m$. I've obviously opted for displaying the variables in the equations in alphabetical order, and it looks wrong because it's against convention - but why are the variables ordered in the way they conventionally are in the first place? From what I can see of all the examples I've just given, the out-of-place variable is _mass_ , so perhaps the convention is to put mass first? I don't see any reason why that would be the case though.


title: A reference frame must be non-moving?

text: I'm tough in school that a reference frame must be non-moving; For example if I take as reference frame the waves of the ocean, i will have the impression that i'm moving, but I'm not. But if movement is relative, how to distinguish between a reference frame that is not moving and one that is moving? It is impossible i think!


Is it reasonable to assume that de Sitter temperature will stop growing when it reaches equilibrium with the temperature of CMB? Currently the de Sitter temperature (the temperature of the Universe's boundary) is $$T_{dS}=2.67\times10^{-30}\,\mathrm K$$ and rises. At the same time, the temperature of CMB is $$T_{cmb}=2.7\,\mathrm K$$ and falls. I wonder, does it mean that at the moment they will equate, the de Sitter temperature will stop growth, and so the expansion of the universe will also stop accelerating, thus reaching the thermodynamic equilibrium? In other words, whether the excessive temperature of CMB compared to the de Sitter temperature is the main source of the accelerated cosmic expansion?


title: Is time travel different from time dilation?

text: I just wanted to know whether time dilation is different from time travel or not. Personally I believe both are different. If they are the same things I just wanted to know why. According to me time dilation is something that applies for every object, while time travel particularly applies for humans. Moreover I feel that travelling through time also in a sense means being able to control time. Time dilation is just an effect due to special relativity and I think it is often confused with time travel. Time travel can be based upon time dilation but I do not think that it completely implies the same thing. **EDIT** By time travel I mean like exact travelling. Like you can go in any direction. Similarly I mean into the past as well as the future. But I just wanted to know if there is a general difference since everyone could define time travel in a separate manner.


title: Superposition and density matrix. What are these states?

text: I just wanted to understand the following. Let's stay with the harmonic oscillator in QM, just to have an example at hand. First, there are all the different states for $n=1,2,...$. (Let's call them $\psi_n$). Then, the superposition of a state, for example $$\frac{1}{\sqrt{2}}(\psi_1+ \psi_2)$$ is also a solution to the Schrödinqer equation. But then, there is also the concept of a density matrix, for example $$\rho= \frac{1}{2} |\psi_1 \rangle \langle \psi_1| + \frac{1}{2} |\psi_2 \rangle \langle \psi_2|.$$ My question is: What is the meaning of the concept of superposition and this density matrix state? What I know so far is somewhat vague:I would say I can only measure whether a particle is in one of the $\psi_i$. The density matrix tells me that the particle is equally likely in one of the two states written down there, but if I would measure, then I would get one of them. But I don't really know what the superposition tells me? Which experimental setting corresponds to a superposition and what is the meaning of it? If anything is unclear about my question, please let me know.


title: interaction between mathematical structures

text: From a physicist's perspective there are several situations in which somehow arbitrary choices of mathematical structures can be made. One can describe a system from different perspectives, etc. without changing anything in the physical properties. (see gauge invariance, picture changing etc.) In the situation when these choices impose a mathematical structure one has limitations in the subsequent choices (are these limitations physical???). However, there are situations when this prescription does not hold. The best example is in this case the Black-Hole complementarity principle (if it is well defined in the first place). So, the question about how mathematical structures "interact" in a logical way is not a bad one. Again: what is the interaction between, say, locality and Hausdorff-ity in the case of a black hole? Or what is the true relation between metrizability and hausdorff-ity of a space in the context of a black hole? The question about how mathematical structures assigned (innocently) at some point interact in different situations is in my opinion a quite relevant one...


title: Double connectivity of $SO(3)$ group manifold

text: Is there any physical significance of the fact that the group manifold (parameter space) of $SO(3)$ is doubly connected? EDIT 1: Let me clarify my question. It was too vague. There exists two equivalence classes of paths in the group manifold of SO(3) or in other words, $\Pi_1(SO(3))=Z_2$. This space is therefore doubly connected. There are paths which come back to initial configurations after a rotation of $2\pi$ and others after a rotation of $4\pi$, with proper parametrization of angles. **Using this fact** , is it possible to show that such a topology admits the existence of half-integer spins and integer spins? I understand spinors as objects **whose wavefunctions** pick up a -ve sign after a rotation of $2\pi$, and comes back to itself after a rotation of $4\pi$. Right? But from the topological argument given above, it is not clear to me, that how does it lead to two kinds of **wavefunctions** , spinor-type $(j=\frac{1}{2},\frac{3}{2},\frac{5}{2}...)$ and tensor-type $j=0,1,2,...$? It is not explicitly clear _how these two types of paths in SO(3) group manifold will lead to such transformation properties on "the wavefunctions"_? EDIT 2: http://en.wikipedia.org/wiki/SO%283%29#Topology.


title: Faradays Law: How Does Curl E Change Over Time In a Coil?

text: I think I am currently making a mistake regarding my interpretation of Faraday's Law. $\vec{\nabla}\times\vec{E}=-\frac{\partial\vec{B}}{\partial t}$ Assuming we have an electric field $\vec{E}$ in the plane between 2 particles given by: $\vec{E}=(\frac{x}{x^2+y^2}+\frac{1-x}{(x-1)^2+y^2}, \frac{y}{x^2+y^2}-\frac{y}{(x-1)^2+y^2})$ Then $\vec{\nabla}\times\vec{E}=(0,0,\frac{y\cdot(2\cdot x-2)}{((x-1)^2+y^2)^2}+\frac{(2\cdot(1-x))\cdot y}{((x-1)^2+y^2)^2}) = (0,0,0)$ As expected, now, if there's an electric field through a coil due to potential difference, then why will the curl be non-zero only when the electric field is initiated and closed off from the coil (e.g. connected/disconnected from a power source)? Currently; I assume that the derivative of the magnetic field converges to 0 over time - because it can not grow indefinitely if we simply have a potential difference (coil connected to a power source). So my question is: How does the electric field (I am correct in that E indeed stands for the electric field, right?) change over time in the time we connect the coil (get potentials at both ends, and a current), until the coil's E curl has stabilized (becomes 0)? It'd be great if someone had an example of an $\vec{E}(\vec{r}, t)$ that changes with time so that $\vec{\nabla}\times\vec{E}$ converges to 0. Where $t_0$ is the connection time and $t_1$ the stabilization time, and $\vec{r}$ the position in space. Are my questions based on false premises? I feel really confused, I've tried my best to look for answers but I can not seem to find them.


title: How we know this solution is unique?

text: A typical problem where we exploit the uniqueness theorem towards a solution, is finding the potential outside two cylindrical parallel conducting wires (at potentials $V_0$ and $-V_0$) extending to infinity. (For example, see Griffiths, 4th ed, prob. 3.12) We obtain a solution that consists of the sum of the potentials of two linear charge distributions lying inside the wires. But why is this solution unique, and how did we know? The hypotheses of the uniqueness theorems stated and proved in the same book don't seem to be satisfied. EDIT: Let us take a simpler case: an infinite grounded conductor in the $xy$-plane with a point charge above it. Even this doesn't seem to satisfy the theorems


title: How to calculate the Wald functional?

text: I want to calculate the Wald functional for arbitrary higher curvature Lagrangians - like getting equation 6.31 from 6.30 in this paper. A priori the above looks like an extremely complicated calculation! Or is there a trick or a software possibly used to get this?


title: Is there a Newton's third law for the em field?

text: There is a momentum associated with the em field that ensures the conservation of total momentum for a system of interacting charges. Can the same be done in an analagous way to ensure Newton's third law is also true?


title: Is it possible to measure the radius of an elementary particle?

text: The only way to describe the electron radius that I found in literature is the "classical electron radius". Is it possible to experimentally measure this? Is there a better way to describe the electron size (and in general, the size of any other elementary particle)?


title: Interpretation of Dirac equation states

text: In Pauli theory the components of two-component wavefunction were interpreted as probability amplitudes of finding the particle in particular spin state. This seems easy to understand. But when talking about Dirac equation, we have four-component wavefunction, two of which correspond to usual spin components of Pauli electron, and another two... How do I interpret positron-related components of Dirac electron? Are they probability amplitudes for the particle to appear to be positron? Or maybe to appear to _not_ be positron (taking Dirac sea picture into account)?


title: If you stop water from expanding, will its temperature stop rising?

text: Water expands when it heats up. If you heat water in a container that prevents it from expanding, will its temperature top out -- maybe around the boiling point? And if not, will it still turn to gas without room to expand?


title: At what size will self-gravitation contribute more to stability than surface tension?

text: The governments of Earth have embarked on an experiment to place a massive ball of water in orbit. (umm... special water that doesn't freeze) Imagine this to be a fluid with a given density, $\rho$ ($kg/m^3$), surface tension, $\sigma$ ($J/m^2$), and formed in a sphere of radius $R$ ($m$). I think that the viscosity $\mu$ is not needed for this question, but correct me if I'm wrong. At what size will the restorative forces from gravity (after some small perturbation) become more significant than that from surface tension? Would the type of perturbation make a difference? Just for fun, here's a video of a ball of water stabilized by surface tension.


title: What is a antiunitary operator?

text: In field theory one can define a time reversal operator T such that $T^{-1} \phi (x) T = \phi (\mathcal T x)$. It is then proved that T must be antiunitary: $T^{-1} i T = -i$. How is this equation to be understood? If $i$ is just the unit complex number, why don't we have $T^{-1} i T = i T^{-1} T$ which is just the identity times $i$?


title: Addition of spin angular momentum for massless particles

text: How do I add the spin angular momentum of massless particles, like photons, where only the transverse polarizations are allowed? If all three polarizations were allowed, this would be an easy exercise: you'd get $S=0,1,2$ states. But some of these states clearly aren't allowed for purely transverse photons: For example, with two photons moving in the same direction as quantization axis, the state $$\big|S=2,M=1\big\rangle=\frac{1}{\sqrt{2}}\big(|m_1=1,m_2=0\rangle+|m_1=0,m_2=1\rangle)$$ contains the $m=0$ state which is not allowed. **Edit** : In the spectroscopic notation, which states are allowed?


title: What's the relation or difference between Lippmann-Schwinger equation and Dyson equation?

text: In quantum scattering theory, Green's Function is defined as [1] $$G_0(z)=(z-H_0)^{-1},$$ $$G(z)=(z-H)^{-1},$$ where $H_0$ and $H=H_0+V$ are separately non-interacting and interacting Hamiltonian. $V$ is interaction.   One can then use the identity $$\tag{1}V=G_0^{-1}-G^{-1},$$ to obtain Lippmann-Schwinger equation $$\tag{2}G=G_0 + G_0 V G. $$ However, on the other hand, in quantum field theory(QFT), Green's function is defined as correlation function. For 2-point Green's function, we have Dyson equation $$\tag{3} G=G_0+ G_0 \Sigma G, $$ where $\Sigma$ is here defined as self-energy. Equivalently $$\tag{4} \Sigma:=G_0^{-1}-G^{-1}.$$ **My questions are**   Are the two Green's functions the same? What's the relation between the two formalisms? And the relation between Lippmann-Schwinger equation and Dyson equation? If they are actually the same thing, then does it mean $V=\Sigma$(this sounds very stupid)? Are the possible differences relating to the discrepancy between S-matrix theory and QFT? [1]: John R. Taylor, Scattering Theory: The Quantum Theory of Nonrelativistic Collisions.


Is there anyway to use a scientific instrument to measure the density of electron around the atomic orbital? Please list both old way and more modern ways.


title: What kind of a particle has this mass?

text: I have a particle that has a mass around $(760\pm10)~MeV/c^2$ but I do not know what kind of particle it is. This links me to some tables that have data on all sorts of subatomic particles but it is a bit daunting to go through and figure out what kind of a particle it could be. Does anyone know off the top of their head what kind of a particle this could be or what class of particle (baryon, meson, specific family of meson, etc.) it could be, or if there is a more effective way than sorting through LBL tables to figure out what kind of particle it is?


title: Is there is a reason for Pauli's Exclusion Principle?

text: As a starting quantum physicist I am very interested in reasons why does Pauli's Exclusion Principle works. I mean standard explanations are not quite satisfying. Of course we can say that is because of fermionic nature of electrons - but it is just the different way to say the same thing. We can say that we need to antisymmetrize the quantum wavefunction for many electrons - well, another different way to say the same. We can say that it is because spin 1/2 of electron - but the hell, fermions has by the definition half- integral spin so it doesn't explain anything. Is the Exclusion Principle something deeper, for example in Dirac's Equation, like spin of the electron? I think it would be satisfying.


title: current electricity and voltaic cell

text: let us consider that i have a voltaic cell in an open circuit that has copper plate at higher potential w.r.t zinc plate. Now my query is, since the electrolyte separating them is a conducting medium, the positive and negative charges on the electrodes must get neutralised immediately(i think so)...But it doesnt happen as i assume? What actually might be my failure in understanding....i do think m ryte....help!


title: Vector cross product of $\mathbf{r}$ and $\ddot{\mathbf{r}}$ in polar coordinates

text: I'm struggling with the following question: > _**Question 6** A planet of mass $m$ moves under the gravitational > attraction of a central star of mass $M$. The equation of motion of the > planet is_ > > $$\ddot{\mathbf{r}} = -\mathcal{G}\frac{(M + m)}{r^3}\mathbf{r}$$ > > _where $\mathbf{r}$ is the position vector of the planet with respect to the > star, $r = \lvert\mathbf{r}\rvert$ is the magnitude of $\mathbf{r}$ and > $\mathcal{G}$ is the universal gravitational constant._ > > _(a) Take the vector product of $\mathbf{r}$ with the above equation for > $\ddot{\mathbf{r}}$ and use the standard result, $\dot{\mathbf{r}} = > \dot{r}\hat{\mathbf{r}} + r\dot{\theta}\hat{\mathbf{\theta}}$ for motion in > a polar coordinate system to show that $r^2\dot{\theta} = h$ where $h$ is a > constant._ Part (a) asks to take vector product of $\mathbf{r}$ and $\ddot{\mathbf{r}}$. I don't know how to do this in polar coordinates.


title: A light flickers at a fast rate why do humans only see light on?

text: A light switches on and off at a fast rate, with equal time off and on, why do we see a light that appears only on rather than only off?


title: Penetration of light in the atmosphere

text: While I was considering an answer to this question, I wondered how much light that enters the atmosphere reaches the ground without colliding with air molecules—if any. I've taken a good bit of physics course (not optics, yet), but I'm not really sure how light interacts with gases or other transparent material. I've always thought that it's periodically absorbed and then reemitted at a similar or identical angle and energy. I know I'll learn about it at some point, and I'll probably read up on it sooner than that, but to make it more fun, I'll ask these questions first: What fraction of the light that enters the earth's atmosphere reaches the ground "unmolested"—without ever colliding with an air molecule? Is there practically any? If no light can traverse the atmosphere without interacting with air, then what fraction of it reaches the ground without significant changes to its direction and/or energy? I imagine that this has a frequency dependency (i.e. the sky is blue, UV filtering). Well, there you have it. Now I'm going to try to focus on my real homework.


title: Quantum entanglement, principle of locality and speed of light

text: I know it's a newbie question but I was wondering about how the principle of locality and the speed of light's limit fit with the phenomena of quantum entanglement? I've read that, due to Bell's theorem, the quantum entanglement does not violate the two principle above, but how it is possible if when entanglement kicks in we cannot differentiate between two reference system no matter at how distant they are? I just want to understand what I don't get of the whole.


title: Difference between Monte Carlo and Quantum Monte Carlo methods?

text: What are the differences between Classical Monte Carlo methods and Quantum Monte Carlo methods in condensed matter physics? If one want to study strongly correlated systems with Quantum Monte Carlo method, does he/she need to study Classical Monte Carlo method first? (just like if you want to study quantum mechanics you shall study classical mechanics first?)


title: Can an invisible light source cast shadows?

text: Let's assume that we have a mechanism for producing EM radiation suspended in the air, and that that mechanism itself is invisible to the naked eye (e.g. a microscopic light bulb on a microscopic wire or a suspended molecular reaction giving off energy.) When off, the light source would not be visible. When turned on however, any light cast from the source to the environment around it would also reach the viewer's eye, and identify the source of light. Is it theoretically possible to cause the lit light source to cast shadows on the environment that are visible, but for the source itself to remain invisible? As an example, most 3D graphics programs can create invisible light sources which are only identifiable by the scene they illuminate. I imagine this may be possible with certain environments, such as an environment covered in phosphorescent paint and a black light source, where the light source is not seen but the environment which it illuminates is identifiable. But what about in the general case, when we limit the variables to the source of the light itself, stipulating it must work in a general environment?


title: How is a Rydberg Blockade Radius defined?

text: Rydberg blockade is a phenomena in 3 or more level systems of Rydberg dressed atoms.


title: Superfields and the Inconsistency of regularization by dimensional reduction

text: # **Question:** How can you show the inconsistency of regularization by dimensional reduction in the $\mathcal{N}=1$ superfield approach (without reducing to components)? * * * # **Background and some references:** Regularization by dimensional reduction (DRed) was introduced by Siegel in 1979 and was shortly after seen to be inconsistent Siegel in 1980. Despite this, it is commonly used in supersymmetric calculations since it has most of the advantages of (normal) dimensional regularization (DReg) and (naively) preserves supersymmetry. The demonstration of the inconsistency of DRed is based on the combination of 4-dimensional identities, such as the product of epsilon-tensors $$ \varepsilon^{\mu_1\mu_2\mu_3\mu_4} \varepsilon^{\nu_1\nu_2\nu_3\nu_4} \propto \det\big((g^{\mu_i\nu_j})\big) $$ and the d-dimensional projections of 4-dimensional objects. Details can be found in the references above and below, although the argument is especially clear in Avdeev and Vladimirov 1983. Various proposals have been made on how to consistently use DRed and most involve restrictions on the use of 4-dimensional identities using epsilon- tensors and $\gamma_5$ matrices. (Note that the treatment of $\gamma_5$ in DReg is also a little tricky...). This means we also have to forgo the use of Fierz identities in the gamma-matrix algebra (which is also a strictly 4-dimensional thing - or whatever integer dimension you're working in). This means we lose most of the advantages that made DRed attractive in the first place - maintaining only the fact that it's better than DReg in SUSY theories. The latest such attempt is Stockinger 2005, but it's also worth looking at the earlier discussions of Delbourgo and Jarvis 1980, Bonneau 1980 and (especially) Avdeev and Vladimirov 1983 & Avdeev and Kamenshchik 1983. The pragmatic discussion in Jack and Jones 1997 is also worth reading - it also contains a fairly complete set of references. Anyway, all of the "fixes" are hard to do when using superfields, since the $D$-algebra has all of the "bad" 4-dimensional algebra built in. **My question is:** What is the easiest way of showing the inconsistency of DRed in the superfield approach? (I want an answer that does not rely on reducing to components!). I'm guessing that it should somehow follow from the $D$-algebra acting on dimensionally reduced superfields.


title: Annihilation and creation operator - $\phi$ and $\pi$ for Klein-Gordon Field

text: **Introduction and Notation** Let $\phi(\vec{x})$ be the real Klein-Gordon (quantum) field, written as: $$\phi(\vec{x})=\int\frac{d^3p}{(2\pi)^3}\frac{1}{\sqrt{2\omega_{p}}}\left(a_{\vec{p}}+a^{\dagger}_{-\vec{p}} \right)e^{i\vec{p}\cdot\vec{x}} $$ where $\omega_{p}=\sqrt{|\vec{p}|^2+m^2}$, $a_{\vec{p}}$, $a^{\dagger}_{-\vec{p}}$ the annihilation and creation operators, and let $\pi(\vec{x})$ the momentum density conjugate to $\phi(\vec{x})$, given by $$\pi(\vec{x})=\int\frac{d^3p}{(2\pi)^3}(-i)\sqrt{\frac{\omega_{p}}{2}}\left(a_{\vec{p}}-a^{\dagger}_{-\vec{p}} \right)e^{i\vec{p}\cdot\vec{x}} $$ **The question** The only non trivial equal-time commutator is $$[\phi(\vec{x}),\pi(\vec{y})]=i\hbar\delta^{(3)}(\vec{x}-\vec{y}) $$ As the relation between $\phi(\vec{x})$ and the $a,a^{\dagger}$ is linear, and so is between $\pi(\vec{x})$ and them, I'm going to express the commutator obeyed by $a$ and $a^{\dagger}$. I'm failing to derive the inverse Fourier transform $$ \int d^3x'\phi(\vec{x})e^{i\vec{p}'\cdot\vec{x}'}=\int\int d^3x'\frac{d^3p}{(2\pi)^3}\frac{1}{\sqrt{2\omega_{p}}}\left(a_{\vec{p}}+a^{\dagger}_{-\vec{p}} \right)e^{i\vec{p}\cdot\vec{x}}e^{i\vec{p}'\cdot\vec{x}'}=\\\=\int d^3x'e^{i\vec{p}'\cdot\vec{x}'} \int\frac{d^3p}{(2\pi)^3}\frac{1}{\sqrt{2\omega_{p}}}\left(a_{\vec{p}}+a^{\dagger}_{-\vec{p}} \right)e^{i\vec{p}\cdot\vec{x}}=\\\=(2\pi)^3\delta^{3}(\vec{p}')\int\frac{d^3p}{(2\pi)^3}\frac{1}{\sqrt{2\omega_{p}}}\left(a_{\vec{p}}+a^{\dagger}_{-\vec{p}} \right)e^{i\vec{p}\cdot\vec{x}}$$ **What I want is a Dirac delta $\delta^{3}(\vec{p}-\vec{p}')$** , so where is my procedure failing? I know this is a math question, but given the physical context, it may fit here better. Thanks for your time, any hint will be appreciated **EDIT 1** **EDIT 2** EDIT 1 Now as an answer


title: Stark effect in two level atomic, laser-driven system?

text: In a two level atomic system (Rabi oscillating problem) the perturbative potential is oscillating / sinusoidal because it comes from the electric field from the laser. Now stark effect is the phenomena of splitting of energy levels in atomic system due to electric field. Here how does the splitting takes place? What is the signature of Stark effect?


title: Radiation at plate of vacuum tube

text: Say you have a vacuum tube, such as the kind used in old amplifiers, wherein electrons are accelerated from the cathode to the anode through an electric field. Presuming, for the sake of argument, that electrons coming off of the cathode have a zero energy, and the anode, made of a material with a work function of 4eV, is at a potential of +24V relative to the cathode, then wouldn't a photon with an energy of 28eV be emitted upon impact of the electron on the anode? Using E=h*f and c=lambda*f, then wouldn't the wavelength of this emitted photon be ~44.28nm? That's beyond the highest reaches of ultraviolet light and into X-ray territory, so why are vacuum tubes safe? Is there something wrong with my understanding of how all this electron/photon business works? Second side question if you care to answer it: I turn off the filament in my vacuum tube and let the cathode get cold, but I keep it plugged into the circuit, with the anode still at +24V relative to cathode, then I shine a light at 635nm on the cathode. Supposing my cathode has a work function of 4eV as well, do electrons hop off of the cathode due to photoelectric effect or not? Without the anode there, I would say no, because h*f-4eV is negative (-2.05eV), meaning the electron doesn't have enough energy to escape the metal plate. With the anode however, there's an electric field pulling the electron as well, so can it make the jump? I calculate it would reach the anode with 25.95eV of energy if it did.


title: Online course on theoretical electrodynamics

text: I'm looking for good online course for an introduction into theoretical electrodynamics. However, it seems that the MIT opencourseware only contains partial lectures for this topic. Has anyone got a recommendation for me?


title: Change of basis in Dirac Notation

text: > `Question`: An operator $A$ is in a particular basis $|a_i\rangle$ (where > $i=1,2$), and is represented by $$A=\begin{pmatrix} 0 & -i \\\ i &0 > \end{pmatrix}$$ Now, define two new basis vectors $|b_i\rangle$ by $$\langle > a_i | b_1\rangle =\frac{1}{\sqrt{2}}\begin{pmatrix} 1 \\\ 1 \end{pmatrix}$$ > $$\langle a_i | b_2\rangle =-\frac{i}{\sqrt{2}}\begin{pmatrix} 1 \\\ -1 > \end{pmatrix}$$ > > What is $A$ in the new basis? `Attempt`: First, I defined the transformation matrix $U$: $$U=\begin{pmatrix} \langle a_1 |b_1 \rangle & \langle a_1 |b_2 \rangle \\\ \langle a_2 |b_1 \rangle & \langle a_2 |b_2 \rangle \end{pmatrix}=\frac{1}{\sqrt{2}}\begin{pmatrix} 1 & -i \\\ 1 & i \end{pmatrix}$$ If we let $A'$ be the matrix in the new basis, we obtain $$ A'=UAU^\dagger$$ And it is a simple calculation (by hand or through mathematica) to show that $$ A'=\begin{pmatrix} 1& 0 \\\ 0 & -1 \end{pmatrix}$$ However, I wanted to check my work using Dirac notation, so I used the equation $$ A_{ij}'=\langle b_i |A|b_j\rangle=\sum_{n,m} \langle b_i |a_n\rangle A_{nm} \langle a_m | b_j \rangle$$ A sample calculation of $A_{12}'$ is shown below: $$A_{12}'=\sum_{nm}\langle b_1 | a_n \rangle A_{nm} \langle a_m | b_2 \rangle$$ $$=\langle b_1|a_1\rangle A_{12}\langle a_2 | b_2 \rangle+ \langle b_1 |a_2\rangle A_{21}\langle a_1|b_2\rangle$$ $$=\frac{1}{2}(1)(-i)(i)+\frac{1}{2}(1)(i)(-i)=1$$ Using the same method for the other components, we find $A'$ to be $$ A'=\begin{pmatrix} 0 & 1 \\\ 1 & 0\end{pmatrix}$$ My question is which method is incorrect? Do I have the formula for the matrix U wrong, or am I not using Dirac notation right? Thank you in advance.


title: What is responsible for cutting an apple: pressure or force?

text: To cut an apple, do I need more pressure or more force? When we want to destroy the intermolecular bonds, don't we need more force ?


title: Rank $L$ spherical harmonic tensor as a $2L+1$ dimensional Cartesian vector?

text: Rank two Cartesian tensors can be decomposed into $L=0,1,2$ spin like things 3x3=1+3+5. But the second equation below does not transform like a "tensor", it looks more like a vector transform in five dimensional Cartesian coordinate. $$\mathfrak{D}^{+}(R)V_{i}\mathfrak{D}(R) = \sum_{j=1,2,3}R_{ij}V_{j}\tag{1} $$ (three dimensional vector.) $$\mathfrak{D}(R)T_{i}^{(2)}\mathfrak{D}^{+}(R) = \sum_{j=-2,-1,0,1,2}\mathfrak{D}_{ij}^{(2)}T_{j}^{(2)}\tag{2}$$ (rank two spherical harmonic tensor.) $$ \mathfrak{B}^{+}(F)V_{i}\mathfrak{B}(F) = \sum_{j=1,2,3,4,5}F_{ij}V_{j}\tag{3}$$ (a vector transform in 5 dimensional Cartesian space.) $(2)$ and $(3)$ above looks the same. My question is can 3d spherical tensors be regarded as high dimensional vectors? Are there some examples?


title: Can there be energy with no force or energy with no power?

text: I think that both force (number of newtons) and power (p=ui(?)) implies that there is energy so we can't have force without energy and we can't have power without energy(?) But can there be energy that is energy and no force and no power?


title: Do all of our discoveries of black holes in nature depend on the validity of GR?

text: In the question Is there a black hole in the centre of the Milky Way? the answer by Motl _seems_ to all but say the existence of that black hole is a fact (see also Evidence for black hole event horizons), whereas the answer by Bunn says "There's a strong consensus among astrophysicists". According to the Chandra X-ray Observatory FAQ the existence of that black hole is conditioned upon the validity of general relativity (GR); the answer there concludes "So, unless Einstein's theory of gravity breaks down, our galactic center must contain a black hole." Questions are:   1. Does humankind's discovery of a black hole in the center of the Milky Way depend on the validity of GR? (If no, can you say why the Chandra X-ray Observatory FAQ contradicts that?) If yes:   2. Do _all_ of our discoveries of black holes in nature depend on the validity of GR? (If no, can you please elaborate on how one was discovered in a way that didn't depend on GR?) **Before answering please note** : I'm not challenging GR in any way whatsoever here. I've tried to phrase the main questions so they can have unambiguous "yes" or "no" answers. As far as I can tell my questions aren't clearly answered elsewhere on this site. I realize that it would be difficult to discover a black hole without in any way depending on a theory of gravity, but I'm asking about a _particular_ theory of gravity here, a theory that predicts black holes.


title: If there's a light ray and it's turned to a new location by a certain angle

text: Imagine that there's a light ray, with source at point **A** , and it's directed towards point **B** (which is very far from point **A** ) and it continues for a huge distance. How will an observer at point **B** perceive the light ray if it gets moved be a certain angle so that it's now directed towards point **C**. If the distance **AB** is large enough, then according to theory of special relativity "something should happen" either with light ray (will it behave like a stream of water, i.e. like matter = matter at the 'end' of stream will fall behind the matter at the 'start') or with perception of the observer so that the speed of light is not exceeded.


title: Multiverse theory and infinite individuals

text: I read an article about possibility of existence of multiverse and came up with a conflicting view with one of the sentences written in the article which goes as follows: > "If space-time goes on forever, then it must start repeating at some point, > because there are a finite number of ways particles can be arranged in space > and time." What does this mean? What does it mean to say when it hints that it must start repeating at some point? Also, where are these multiverses enclosed within? If they are expanding at such a rate, is the container in which they are enclosed also expanding?


title: Gas transformation for different molar number

text: When a gas at normal conditions ( **1atm, 273K, 22.4L** ) we say like the $$\frac{PV}{T} = 0.0820...$$ And by the know equation: $$PV = nRT$$ Where R is equals to $0.08205...$ and $n$ is the number of mols of the substance. When we want to compare a previous state of a gas with its next state, we use the formula: $$\frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2}$$ But the $R$ is just the first state of the gas, already in a constant term ($\frac{P_1 V_1}{T_1}$ is $\frac{1 \cdot 22.4}{273}$ that is $R=0.082...$) So, instead of doing: $$\frac{1 \cdot 22.4}{273} = \frac{P_2 V_2}{T_2}$$ We do: $$R = \frac{P_2 V_2}{T_2}\tag{assuming the same n of mols}$$ But let's say we're doing this equality for a gas that has a given number $n$ mols of the substance. By what I know it is $$nR = \frac{P_2 V_2}{T_2}$$ I have $2$ questions: $(1)$ - How do I know that for a $1/2$ number of mols (for example), the quotient $\frac{PV}{T}$ is gonna be exactly $1/2$ (in other words, how do I know that they are linear) $(2)$ - Why do I have to multiply the $n$ always for the $\frac{1 \cdot 22.4}{273}$ and not for the $ \frac{P_2 V_2}{T_2}$ in the equation? (in other words, if I'm just equating two states of the gas, I guess there should be no problem with which side I multiply by $n$). I don't get what "multiplying by $n$" means. Like, the two states of the gas that I'm equating have the same amount of molecules, doesn't mean if they are $1/2$ mol or anything, they're the same quantity. Thanks!


title: How to directly calculate the infinitesimal generator of SU(2)

text: We commonly investigate the properties of SU(2) on the basis of SO(3). However, I want to directly calculte the infinitesimal generator of SU(2) according to the definition $$X_{i}=\frac{\partial U}{\partial \alpha_{i}}$$ from Lie group theory. But, where are the problems of the methods I used below? First, I parameterize the SU(2) with the $(\theta, \phi, \gamma) $ like this: $$U=\begin{bmatrix} e^{i\theta}sin\phi & e^{i\gamma}cos\phi \\\ -e^{-i\gamma}cos\phi & e^{-i\theta}sin\phi\end{bmatrix}$$ and the E is when $(\theta, \phi, \gamma)$= $(0,\frac{\pi}{2},0)$. Second, I use the definition of infinitesimal generator like this: $$ I_{1}=\frac{\partial U}{\partial \theta}|_{(0,\frac{\pi}{2},0)}=i\begin{bmatrix}1 & 0\\\ 0 & -1 \end{bmatrix}$$ $$ I_{2}=\frac{\partial U}{\partial \phi}|_{(0,\frac{\pi}{2},0)}=i\begin{bmatrix}0 & i\\\ -i & 0 \end{bmatrix}$$ $$ I_{3}=\frac{\partial U}{\partial \gamma}|_{(0,\frac{\pi}{2},0)}=i\begin{bmatrix}0 & 0\\\ 0 & 0 \end{bmatrix}$$ Here is the question... Why do I get the 0 matrix? We should expect to have the Pauli Matrix. Isn't it?   Where is the problem from?


title: Electromagnetic field to cool a substance?

text: I saw somewhere that an electromagnetic field would cause a substance to let off thermal energy, ultimately resulting in the substance to cool really quickly. If this is possible, does the strength of the magnetic field correlate with the rate of cooling? What are the physics behind this phenomena?


title: How fast will I get if I burn a comet?

text: Assuming I have a spacecraft which is $30,000\,\mathrm{kg}$ (roughly the size of the Apollo spacecraft). If I take a comet and (theoretically) electrolize it perfectly to hydrogen and oxygen. I know have $2.2\times10^{14}\,\mathrm{kg}$ of $\mathrm{H_2}$ and $\mathrm{O_2}$. Now I burn it. Assuming perfect energy consumption (all released energy goes to propulsion) how fast will my spacecraft get? On one hand you have the $F=\frac{GM_1M_2}{R^2}$ which decreases due to both $R$ growing and $M_1$ (the mass of rocket+propulsion) decreasing. How fast will this spacecraft get?


title: What is the fastest a spacecraft can get using gravity-assist?

text: Assuming normal spacecraft and space objects (no neutron stars, black holes, etc). To what speed can a spacecraft accelerate using gravity-assist? For example, if a spacecraft is moving at relativistic speeds, it probably won't get seriously sped up by normal-density objects.


title: mirror house gedanken experiment

text: Imagine a mirror house i.e. completely made of mirror with no traces of whatsoever light absorbent. Now, you introduce a light beam into that room and observe somehow through a hole. Would the room be   1. complete dark - since nothing is actually absorbing it!! or would it be   2. very bright - since light is getting reflected from every place possible? What I want to compare the above situation is with a white washed room. * * * **EDIT:::Clarifications** This question is the result of a debate with my friend, who says the room would be dark and me who says otherwise. Now being said that let's move to specifics...let's assume that the room is a cuboid and the light source is say a light bulb i.e. an isotropic source like a typical room at ceiling. and the observer with infinitesimally small view port looking from one of the wall presumably neither the ceiling nor floor. I hope this suffices.


title: How can primordial helium be formed before atoms?

text: In my physics textbook it describes the events at the beginning of the Universe. I'm confused about the order at a certain point. It says that at some point primordial helium is created, then it says that later atoms are formed. Isn't primordial helium made of atoms? Thank you :)


title: Why does Hydrogen molar heat capacity reach 7/2 R?

text: If a diatomic gas like Hydrogen has 6 maximum degrees of freedom why its molar heat capacity reaches at high temperatures $C_V = \frac{7}{2} R$ and not $C_V = \frac{6}{2} R= 3R$? molar heat capacity of hydrogen


title: The Role of Rigor

text: The purpose of this question is to ask about the role of mathematical rigor in physics. In order to formulate a question that can be answered, and not just discussed, I divided this large issue into five specific questions. 1) What are the most important and the oldest insights (notions, results) from physics that are still lacking rigorous mathematical formulation/proofs. 2) The endeavor of rigorous mathematical explanations, formulations, and proofs for notions and results from physics is mainly taken by mathematicians. What are examples that this endeavor was beneficial to physics itself. 3) What are examples that insisting on rigour delayed progress in physics. 4) What are examples that solid mathematical understanding of certain issues from physics came from further developements in physics itself. (In particular, I am interested in cases where mathematical rigorous understanding of issues from classical mechanics required quantum mechenics, and also in cases where progress in physics was crucial to rigorous mathematical solutions of questions in mathematics not originated in physics.) 5) The role of rogor is intensly discussed in popular books and blogs. Please supply references (or better annotated references) to academic studies of the role of mathematical rigour in modern physics. (Of course, I will be also thankful to answers which elaborate on a single item related to a single question out of these five questions.) Related Math Overflow questions: Examples-of-non-rigorous-but-efficient- mathematical-methods-in-physics (related to question 1); Examples-of-using- physical-intuition-to-solve-math-problems; Demonstrating-that-rigour-is- important


title: I need a micro-sized clutch for a project, what are my options?

text: Also, what's the proper stack exchange site to ask this on? mechanics.stackexchange seems to be for motorvehicles. I'm designing an automatic guitar tuner that clamps onto my acoustic guitar. The tuning key handles will be removed and replaced with one that has a brass gear on it. Each tuning key gear is then turned by its own micro-motor gearbox, so there are a total of 6 on a 6-string guitar. The problem is that I want the user to still be able to turn the tuning key handles manually, so that manual tuning of the guitar is still an option. So I thought 'oh, a ratchet clutch...' but that would only allow one direction of rotation and I need either direction (for tuning to lower or higher pitch). So then I thought 'a slip clutch'. Well what are my other options, and will I need to manufacture this micro-motor clutch myself? The force needed to slip the clutch will be greater than the force needed to turn the tuning key obviously, and less than the force which would break gears in the micro-motor gearbox. Thanks.


title: Viewing glass from an oblique angle

text: When I view most glass from the side it's green which I've found out is due to impurities in the glass specifically from iron oxide. Why is it when I view the larger face from an oblique angle, it isn't nearly as green?      I cannot personally notice any different on the piece I have next to me even when I hold it at an angle that would be almost looking at the edge of the glass. It is pretty small (about 2.5" x 5" x .0625" or about 61mm x 127mm x 2mm, l x w x h) but I feel like it's big enough that I'd be looking through enough glass to get the green.


title: Determinism loophole?

text: I was thinking about the question I posted yesterday, and I thought of a better way to ask it. I'm trying to figure out why QM necessitates "pure randomness". Assume you have a photon that has a hidden variable. This hidden variable is a pseudorandom number generator $f(t) \in \mathbb{R}$ such that $0 \leq f(t) \leq 1$. If $f(t) > 0.5$, the photon passes through the polarizer, and if $f(t) \leq 0.5$, it does not. If the experimenter could figure out what this PRNG is, he could predict the result of every measurement, which is more than QM can predict. In other words, the photon has a local hidden variable that if known would remove the possibility of "true" randomness, while still reproducing the probability distribution predicted by QM. However, Bell's theorem rules out this possibility. That's not what I have a problem with — giving up locality is fine with me. So consider this: The PRNG is no longer a hidden variable of each photon, but a hidden variable of an entangled two photon system. I'm sure this can be done with one PRNG, but for simplicity of explanation, let's say there are two individual PRNG's associated with the entire system: $g_1(t)$ and $g_2(t)$. The photons are entangled and separated. Photon 1 heads toward polarizer 1 with angle $\theta_1$ and photon 2 heads toward polarizer 2 with angle $\theta_2$. It's well known that the probability that each photon gives the same measurement is given by: $$P(\theta_1, \theta_2) = \cos^2(\theta_1 - \theta_2)$$ and this has been experimentally verified. It's clear to see that because the angles of each polarizer can be altered _while_ each photon is still in flight, there must be an instantaneous connection between the measurement results. However, to me, this still doesn't imply true randomness. Suppose photon 1 gets to its polarizer first at time $t_1$. Whether it passes through the polarizer is simply given by the boolean $X_1 = g_1(t_1) > 0.5$. Now define another boolean $$Y = g_2(t_2) < \cos^2 (\theta_1 - \theta_2)$$ , where $t_2$ is the time that photon 2 arrives at its polarizer. Whether photon 2 passes through the polarizer is then given by: $$ \overline{X}_1 \overline{Y} + X_1 Y$$ As far as I can tell, this doesn't violate any of the postulates of QM or any kind of no-go theorem, and it's deterministic. Where did I go wrong?


title: Could elementary particles join to create bound states?

text: I recently had question, can 2 elementary particles be "attached" together using Strong and weak nuclear forces to create a bound state. For example can Electron and some other stable elementary particle such as Strange Quark be formed to gather with electron orbiting the heavier particle similar to a atomic structure?


title: Stern Gerlach with spin in opposite directions

text: So for the Stern-Gerlach apparatus, we assume that we either have a particle spin up or spin down. We also have the varying field, $\partial B/\partial z$. This initial configuration results in the particle wither going to plus $\hbar$ or minus $\hbar$. Suppose instead of having spin up/down in the z direction, I sent it through with an initial spin aligned in the x direction (same exact configuration)? The Hamiltonian is given (for a linear B) as $$H=\frac{1}{2m}(p_x^2+p_z^2)-\mu \sigma_z(B_0+B'z)$$ So my equations of motion for the z direction would just give me $p_0t/m+z_0$ and $\dot p_x=0$. Do I need to account for the spin x now instead, or will the particle go undeflected?


title: Mechanics of a rolling drum

text: ![image_of_problem](http://i.stack.imgur.com/pUJyJ.png) I have no clue on how to approach this. The professor only discussed centripetal acceleration and angular velocity (As in $2πr\over T$ $= ωr$). Does the acceleration along the axis of the drum act in the same way that centripetal acceleration does? I understand that the angular velocity in this case is $ωR$, and the derivative of that is the angular acceleration, but how to I find $ω$? How does $a$ affect the period, and how does its position (In this case along the axis) affect that?


title: What is wrong with this reasoning on chemical potential?

text: One definition of the chemical potential is the change of internal energy of the system with respect to a particle of the added substance with the system entropy, molar volume, and all other species particle numbers remaining constant. A thought experiment: Let's suppose I have a completely rigid container with 1 mole of argon in it with an infinite amount of insulation surrounding the container. Therefore, I have zero heat transfer and the molar volume is constant. I admit one atom of argon into the container in a reversible manner (same P and T). The pressure of the system must increase differentially according to the ideal gas law. Hence the entropy will decrease differentially except that it must stay constant and so temperature will have to increase for this process to be isentropic. But for an ideal gas, internal energy depends only on temperature. So dU/dn is positive. Yet, when I look up values for argon as a gas at STP, its value is 0 (like all the elements in their standard states). Where is my reasoning wrong?


title: Time dilation only on electromagnetic force?

text: We've seen by experiment that the speed of light _c_ appears to be constant for each observer (leading to all well-known consequences of relativity). I'm wondering if this appearance of constancy of _c_ might be due to the observer's way of measuring it: All observers are bound to compare _c_ to something else which itself is also based on _c_. A clock based on a photon bouncing between two mirrors (and taking the time it takes to bounce) for instance uses that speed of the photon to measure everything. A clock like a watch based on springs uses tension forces buried in the spring material (electromagnetic forces are based on _c_ ). Quartz crystal oszillators, sand clocks (hourglasses), water clocks — all facilitate some mechanism like friction or piezoelectricity which fundamentally are electromagnetism. Nevertheless it is said that _the time_ appears to be going slower, not just _all clocks we can build_. My questions now are: Is there a reasoning (which I just didn't find in my research) why the _time_ as a whole is supposed to be influenced by relativity, not just all events based on the forces based on _c_? Maybe there even is a word or a term to google for in order to find more about this? I understand that physicists managed to unite three of the four basic forces, wrapping up electromagnetism with the strong and the weak force. I guess then that these additional two forces also are based on _c_. Is there any such connection of _c_ to the remaining force, the gravitation? I could understand that if all existing forces are hinged on _c_ then there is no real difference between saying "all clocks we can build are going slower" and "the time itself is going slower".


title: Increase in cloth's mass

text: When we iron our clothes does the mass of the cloth increase, since energy transformation and transfer happens, Einstein's energy mass relation comes to action and causes increase in mass. Am I right or wrong?


title: Density of Solid States of Compounds

text: One of the wonderful properties of water (as my high school biology teacher would say) is that in its solid form, it is lighter than its liquid form. This means that when temperatures drop below 0 degrees Celsius, the top layer of water on, say, a lake freezes first. This works out pretty well for any fish or other aquatic creatures living underneath, because the lower layers of water will not freeze. I know that this is an exception to the general rule of solid and liquid states: A given substance, when transformed into its solid state, will generally sink in a container of its liquid state. My question is this: What other substances are exceptions to this rule (if any?). What features do they share with water that are responsible for this?


title: Magnetic field, alternated current and EMF

text: If I apply an alternated current to a solenoid and insert into it a smaller solenoid, I could measure the induced EMF (electromotive force) and study how it changes in relation to the frequency of signal generator. If I increase the frequency, I think that the magnetic fiel increases and so also the EMF increases, isn't it? If I insert a metallic cylinder between the two solenoids, the cyclinder "shields" the magnetic field.. but what about EMF? And what about EMF if the cylinder is made by iron?


title: Does sending data down a fiber optic cable take longer if the cable is bent?

text: Ok, so, my simplified understanding of fiber optics is that light is sent down the cable and it rebounds off the sides to end up at its destination. Which got me thinking, if it has to bounce more times (and having a shorter travel between each bounce), does the light (data) take longer to get to the other end of the cable? Like this: http://i.imgur.com/pCHUf.jpg A part of me is saying no, because it's still the same distance to travel and bouncing doesn't take up any time, but another part of me is saying yes because the light will have further to travel the more times it bounces, and thus will take longer to get to its destination. I'm swaying towards it taking more time. Thanks!


title: Experimental proof of gravitational redshift of light

text: Has the gravitational red shift been proven for electromagnetic waves only or also for a single photon?


title: what will happen when I continuously heat a closed heavy iron box half-filled with water?

text: I mean what will be the situation inside? All water vaporized, Equilibrium, Temperature-pressure situation.. boiling point increase/decrease? Thanks for your time.


title: A simple pendulum moving at a relativistic speed - how does the period change?

text: I've been pondering the precise mechanism of time dilation for the example of a simple pendulum in two different situations:   1. The observer and ground are at rest in one frame of reference; the pendulum is moving at high speed with respect to that frame.   2. The observer is at rest in one frame of reference; the pendulum and the ground together move at high speed with respect to that frame. _user8260_ has pointed out that in situation 1, in the pendulum's frame $g$ is greater by $\gamma^2$ compared to $g$ in the observer's frame. Thus in the pendulum's frame the period is less than it is in the observer's frame by a factor of $1/\gamma$, just as one would expect from time dilation. But what about situation 2? Here, compared to the pendulum frame, the observer sees the pendulum with the same length in a stronger gravitational field, yet observes a longer period. Does the inertial mass of the pendulum change differently than its gravitational mass? Also, does the analysis depend on whether the plane of swing is perpendicular or parallel the direction of motion?


title: Proton flux model or data at energy range up to 100 keV

text: I am looking for a model or at best the database of proton fluxes (solar p+) at a energy range of some eV up to 100 keV. I have already found the SOHO database: http://umtof.umd.edu/pm/ But the most energetic p+ was something around 5 keV. Still, I need fluxes from 5 keV up to 100 keV. Is there any database, and maybe a reference to a mathematical model?


title: Reduction of a sum to the first Brillouin zone in a band structure calculation

text: this might be a "standard trick" for many solid state physicists, however it's one that I'm not familiar with so maybe you can help me. Here's the Problem: Suppose we're given a Hamiltonian of the form $H=\sum_{k} \epsilon_{k} c^{\dagger}_{k}c_{k} + \sum_{k} U c^{\dagger}_{k+Q} c_{k}$. Here U is some complex (!) number, k is a 2 dimensional wavevector and Q=$(\pi,\pi)$. Furthermore we impose $\epsilon_k = - \epsilon_{k+Q}$. The first Brillouin Zone is the set $\\{ (k_x,k_y) ; |k_x|+|k_y|<\pi \\} $ in 2D-k-space. Now define the 2D-vector $\Psi_k = (c_k,c_{k+Q})$. Then the Hamiltonian be written as: $H= \sum_{k}' \Psi_k^{\dagger} A_k \Psi_k $ with the k-dependent 2x2 Matrix $A_k$ defined by: $$ \left[ \begin{array}{ c c } \epsilon_k & U \\\ U^* & \epsilon_{k+Q} \end{array} \right] $$ The prime (') in the sum denotes that it has to be taken over wavevectors in the frist Brillouin zone only! Now multiplying this quadratic form out "in reverse" I obtain something like: $H=\sum_{k} \epsilon_{k} c^{\dagger}_{k}c_{k} + \sum_{k} U c^{\dagger}_{k+Q} c_{k} + \sum_{k} \epsilon_{k+Q} c^{\dagger}_{k+Q}c_{k+Q} + \sum_{k} U c^{\dagger}_{k} c_{k+Q} $ It's not clear to me why the third and fourth term are supposed to vanish in case I'm restricting my sum to the frist Brillouin zone. I hope someone can help. It should be rather technical, but still important I think. Thanks in advance.


title: Which is more efficient, heating water in microwave or electric stove?

text: So our propane tank in the kitchen ran out again today. Which is more energy efficient, boiling water in a microwave on an electric stove? All things being equal i.e. starting temperature and mass of water. Not so much about which is faster, but which will cost us less kWh generally. I realize boiling from the stove noticeably heats up the environment as well, and continues emitting warmth long after its power had been switched off. Does the kettle have a higher thermal capacity than the micro-safe glass container (therefore needing to absorb more calories) or is that difference negligible with say 1kg of water? Haven't been inside a microwave to feel its thermal capacity/overhead though. As far as dominant conduction/convection/radiation methods of transfer, it seems fairly obvious in both cases.


title: What is the origin of the naming convention for position functions?

text: In physics, position as a function of time is generally called `d(t)` or `s(t)`. Using "d" is pretty intuitive, however I haven't been able to figure out why "s" is used as well. Is it possibly based on another language?


title: Coulomb splitting in atomic physics

text: In atomic physics, if a configuration with equivalent electrons in some shell (say Neodymium : $[Xe] 6s^24f^4$) gives same $L = \sum_i l_i$ and $S = \sum_i s_i$ ($i$ denoting individual spin and orbital momentum of the electrons, and in this example $L = 1$ appears twice with $S = 0$) considering the Coulomb interaction between electrons ($L$ and $S$ being the good quantum numbers in this case), how can these two $(L,S) = (0,0)$ have different energies ? I don't really see what can split these two levels if they start from the same configuration. Does someone know if they actually split ? And if they do, why ?


title: Helmholtz free energy minimization using Lagrange

text: can you please help me with that one: Minimize free energy for a liquid crystal: $F = \int (K_{11} (div(n))^2 + K_{22} (n*rot(n) + q)^2 + K_{33} (n*rot(n))^2 ) dV$ in the case $n = cos(\alpha)*e_z + sin(\alpha)*e_\phi$ in cylindrical coordinates. $\alpha$ is only dependent of $r$. Goal is to get second order differential equation $\alpha = \alpha(r)$ with the Euler Lagrange law.


title: effects of sunlight through a glass window?

text: Both my father and my grandfather where drivers, and over time ended up with a wrinklier left hand compared to the right hand, due to sunlight exposure over 40+ years while holding the steering wheel with the left hand. I googled for sunlight exposure through a glass window and I get contradictory answers either stating either that most glasses filter out UV-light that affects skin tanning and ageing or that the effect of sunlight on skin is pretty much the same if it is direct or through a glass window. So, are the effects of sunlight through a glass window different from direct sunlight exposure?


title: Could quarks be free in higher-dimensional space than 3D?

text: Reading this answer, I now wonder: if quarks are confined by $r^2$ potential, could their potential allow infinite motion in higher-dimensional space? To understand why I thought this might be possible, see what we have with electrostatic potential: in 3D it is proportional to $r^{-1}$. This is just what Poisson equation tells us for point charge. If we solve Poisson equation in 2D space, we'll see potential is proportional to $\ln\frac r {r_0}$, and in 1D it's proportional to $r$. We can see that it only allows infinite motion starting form 3D. Could the same hold for quarks, but with some higher than 3D dimension? Or is their potential of completely different nature with respect to space dimensionality?


title: Inertial Frames of Reference - Inertial vs. Accelerated Frames

text: According to Robert Resnick's book "Introduction to Special Relativity", a line states the following as the definition of an inertial frame of reference: "We define an inertial system as a frame of reference in which the law of inertia - Newton's first law - holds. In such a system, which we may also describe as an unaccelerated system, a body that is acted on by zero net external force will move with a constant velocity."      Therefore, a frame of reference, with respect to which, objects move in a straight line with uniform velocity in the absence of any unbalanced forces. Now my problem with Resnick's definition arises from the above line: "...In such a system, which we may also describe as an unaccelerated system...". How can an observer, occupying a particular frame of reference, realize that he is part of an unaccelerated system. He can only state, that with respect to other frames of reference, there is a uniform relative motion in a straight line. The definition of an inertial frame of reference is restricted only to comparisons between frames of reference. If a frame of reference is to be considered an inertial one, the condition that its relative motion with respect to other frames of reference should be uniform motion in a straight line, is to be fulfilled. Here is where my confusion arises with relation to the above quoted statement: If, for instance,the relative motion observed between two frames of reference is that of uniform acceleration, how can we determine which frame is the unaccelerated system? It is obviously not possible. But according to the statement made above, Resnick states that the frame of reference he occupies is an unaccelerated one. With respect to what? If accelerated motion were to be observed with respect to other frames of reference, how are we to determine that we occupy an inertial frame of reference at all?      Similarly, another statement made by Resnick in his book, which is related to the above question is as follows: "The objects whose motions we study may be accelerating with respect to such frames but the frames themselves are unaccelerated."      He states that inertial frames of reference are still to be considered frames of reference if other frames of reference are accelerating with respect to the occupied frame of reference. My simple question is this: How can we define an inertial frame of reference as an unaccelerated frame of reference unless and until we observe this particular frame of reference from another frame of reference such that the relative motion between these frames of reference is uniform motion along a straight line as per Newton's first law. Another part of this very question is also: How can we call the occupied frame of reference as being inertial regardless of whether other frames of reference are accelerating with respect to the occupied frame of reference? Please resolve these questions as best as you can without any ambiguity, as you know, specificity is very important in conveying ideas regarding to relativity.


title: Results of Statistical Mechanics first obtained by formal mathematical methods

text: I have a question that seems natural in Physics and Mathematics mainly in Statistical Mechanics of Equilibrium. Results that are proven by formal mathematical methods that were already seem intuitive results and experimentally verified when proofs by mathematically rigorous methods were obtained. In many fields of science, most notably physics, there are many historical examples that refute this view. An example of this type that I know is the General Theory of Relativity and GPS (the original English acronym for Global Positioning System). It is possible that one may disagree. Without explaining in detail the formulas of contraction / dilation of spacetime which is obtained formally Theory of Relativity are used to "sync" properly watch each of the GPS satellites with the clocks of a point on Earth. For more see here and here. However, I can not get an explicit example in Statistical Mechanics. That is, would an example of resutado first obtained by formal mathematical methods and was discovered by experimental means later. **Question:** There is a exemple of same result in Statistical Mechanics that was first discovered by "theoretical mathematical methods" and only later confirmed experimentally ? **Question:** Some example explicit or reference for the Ising model?


title: Calculate displacement in position from knowing constant acceleration

text: I have recently started studying physics at school, and my teacher went over the following equation without explaining about it too much: $$d~=~vt+\frac{1}{2}a t^2.$$ I have wondered, why would this formula actually work? Is there an explanation for this?


title: The string Poisson bracket

text: Where does the factor $\frac{1}{T}$ ($T$ is the string tension) in this Poisson bracket come from? $$ \\{X^{\mu}(\tau,\sigma),\dot{X}^{\nu}(\tau,\sigma')\\} ~=~ \frac{1}{T}\delta(\sigma-\sigma')\eta_{\mu\nu}. $$ I think I can see from remembering the definition of a Poisson bracket (for example in canonical coordinates) why in terms of momentum we have $$ \\{P^{\mu}(\tau,\sigma),X^{\nu}(\tau,\sigma')\\} ~=~ \delta(\sigma-\sigma')\eta_{\mu\nu} $$ but I don't see why this factor in the first equation has to be there. In addition to deriving it by calculation, is there an intuitive physical way how one can see why the factor of inverse tension has to be there, similar to explaining the appearance of the tension in front of the integral in the action by the fact that it costs energy to stretch the world-sheet?


title: Self-adjoint and unbounded operators in QM

text: 1. An operator $A$ is said to be self-adjoint if $(\chi,A\psi)=(A\chi,\psi)$ for $\psi, \chi \in D_A$ and $D_A=D_{A^\dagger}$. But for the free particle momentum operator $\hat{p}$ these inner products does not exist, however its eigenvalues are real. So, is $\hat{p}$ a self-adjoint operator?    2. Why are the operators in quantum mechanics in general unbounded?


title: Interpretation of "transition rate" in Fermi's golden rule

text: This is a question I asked myself a couple of years back, and which a student recently reminded me of. My off-the-cuff answer is wrong, and whilst I can make some hand-waving responses I'd like a canonical one! In the derivation of Fermi's Golden Rule (#2 of course), one first calculates the quantity $P(t)\equiv P_{a\rightarrow b}(t)$ to lowest order in $t$. This is the probability that, if the system was in initial state $a$, and a measurement is made after a time $t$, the system is found to be in state $b$. One finds that, to lowest order in the perturbation and for $\left< a \mid b \right > = 0$, $$P(t) \propto \left( \frac{\sin(\omega t/2)}{\omega/2} \right)^2, \qquad \hbar\omega = E_b - E_a $$ Then one says $P(t) = \text{const.} \times t \times f_\omega(t)$ where **as $t$ increases $f$ becomes very sharply peaked around $\omega=0$, with peak of height $t$ and width $1/t$, and with total area below the curve fixed at $2\pi$**. In other words, $f_t(\omega)$ looks like $2\pi \delta(\omega)$ for large $t$. Now suppose we consider the total probability $Q(t)$ of jumping to any one of a family of interesting states, e.g. emitting photons of arbitrary momenta. Accordingly, let us assume a continuum of states with density in energy given by $\rho(\omega)$. Then one deduces that $Q(t) \sim \text{const.} \times t \rho$, and defines a " **transition rate** " by $Q(t)/t$ which we note is independent of time. The issue I have with this is the following: $Q(t)/t$ has the very specific meaning of "The chance that a jump $a \to F$ (for a family $F$ of interesting states) occurs after making a measurement a time $t$ from the system being in state $a$, divided by the time we wait to make this measurement." It is not immediately clear to me why this is a quantity in a physical/experimental context which is deserving of the name "transition rate". In particular, note that   * $t$ must be large enough that the $\delta$ function approximation is reasonable, so the small-$t$ regime of the formula is not trustworthy;   * $t$ must be small enough that the perturbation expansion is reasonable (and also presumably so that the $\delta$ function approximation is not insanely sensitive to whether there is a genuine continuum of states or simply very finely spaced states) so the large-$t$ regime of the formula is not trustworthy.   * Therefore the physical setup in which one measures $P(t)/t$ events per unit time must use properties as if some measurement/decoherence occurs made in some intermediate range of $t$. **What is the microscopic detail of this physical setup, and why is this intermediate range interesting?**   * _Edit_ : Also I would like to emphasize that the nature of $P(t),Q(t)$ is such that whenever one "makes a measurement", the "time since in initial state" is reset to 0. It seems that the "time between measurements" is in this intermediate range. (Of course, this isn't necessarily about measurements, but might be to do with decoherence times or similar too, I'm simply not sure.) People tell me that the Golden Rule is used in calculating lifetimes on occasion, so I would like to understand why this works! * * * _Succinct question_ : In what sense is $Q(t)/t$ a **transition rate**?


title: What is the physical meaning of the "decay rate" in Fermis golden rule?

text: As far as I understood, Fermi's golden rule gives a prediction of the transition rate in a perturbed quantum system $H_0+V$ between two eigenstates of the unperturbed system $H_0$, say from $\left| i\right>$ to $\left| f\right>$ with eigenenergies $E_i,E_f$. The prediction is perturbative in the first order of $V$. The result is, that the rate equals $$\gamma_{i\to f} = 2\pi |\left<f|V|i\right>|^2 \delta(E_i-E_f)$$ This rate is defined as $$\gamma_{i\to f}=\frac{d}{dt} |\left<f|U(t)|i\right>|^2.$$ **My question:** How should one interpret this differential "rate" physically? **My thoughts so far:** Apparently it is the derivative of the _transition probability_ with respect to time, so it is not a probability itself. In literature it is often called _decay rate_ , but for an exponentially decaying probability $p(t)\sim e^{-\lambda t}$ the decay rate $-\lambda$ would not be computed as $dp/dt$, but $(dp/dt)/p$.


title: relationship between Fredkin gate and the swap and I gates

text: I need to prove this relationship: $$G_{Fredkin} = I \otimes |0\rangle\langle 0| + G_{swap} \otimes |1\rangle\langle 1|$$ with $$G_{swap} = |00\rangle\langle 00| + |01\rangle\langle 10| + |10\rangle\langle 01| + |11\rangle\langle 11|$$ So I think I need to show that both sides are a linear map on $H \otimes H \otimes H$ So what is the basis for $H \otimes H \otimes H$, and then how do I show that $H \otimes H = I$? Is this the right approach?


title: Why does surface charge not move?

text: If you have a wire with current flowing through it, and the current flowing the wrong way (not parallel to the wire) surface charge will buildup, generating a field to force the current to flow the right way. What's stopping the surface charge from redistributing? After the correct distribution has been set up for the surface charge to flow, what's stopping the surface charge from redistributing itself over the conductor to maximize distance between like charges?


title: Two identical particles with spin $s$. What is the spin of its corresponding "center-of-mass" and "relative" particles?

text: Consider a system of two identical quantum particles with spin $s$ and mass $m$. Using center-of-mass coordinates one obtains an equivalent system given by a particle of mass $2m$ and one of mass $m/2$. Can we talk about the $\textit{spin}$ of these two new fictitious particles? What can we say?


title: Relationship between irradiance and radiance

text: A question related to radiometry: Irradiance $E$ at a point $x$ can be written as: $E = \int_\Omega L(x, \omega) cos(\theta) d\omega$ I understand this formula and where it comes from. The equation for radiance can be written as: $L = {d^2\Phi \over {d\omega dA^\perp cos(\theta)}}$ What I don't understand is that if we substitute this last equation in the first one for irradiance, don't we get: $E = \int_\Omega {d^2\Phi \over {d\omega dA^\perp cos(\theta)}} cos(\theta) d\omega \rightarrow \int_\Omega {d^2\Phi \over {dA^\perp}} \rightarrow \int_\Omega d({d\Phi \over dA^\perp}) \rightarrow \int_\Omega dE$ which doesn't make sense to me? What am I missing? Irradiance, integral of differential irradiance over the hemisphere?


title: Where does this energy equation come from: $E_k=k\frac{k(q_1q_2)}{r}$

text: I have this equation but I have no idea where it came from. $E_k=k\frac{(q_1q_2)}{r}$ I don't understand especially since I have this other equation: $E = k\frac{Q_{source}}{r^2}$ And cannot find any relation between the two. Please any helps or suggestions would be useful. Thanks! **EDIT:** I have researched this and either am looking in the wrong place or just can't find it anywhere so please help! :)


title: Shine a light into a superconductor

text: A type-I superconductor can expel almost all magnetic flux (below some critical value $H_c$) from its interior when superconducting. Light as we know is an electromagnetic wave. So what would happen if we shine a light deep into a type-I superconductor? Suppose the magnetic field of the light is significantly lower than $H_c$. And what about type-II superconductors?


title: Molecular spectroscopy for atomic spectroscopers

text: My experience is with atomic spectroscopy of alkali atoms, I've recently been asked by a friend to help with some advice on analyzing molecular spectrophotometry data in the context of molecular biology, To fit spectroscopic data to extract density in atomic spectroscopy you typically need to know the relative transition strengths and natural linewidths for all transitions (for hyperfine resolution spectroscopy), know the interaction path length, know the doppler broadening coefficient (and know the temperature), construct a voigt profile and fit. If I wanted to fit spectroscopic data to extract, say, ATP concentration from a spectrophotometer sweep, would I be able to do this by determining the corresponding molecular parameters? Are there caveats to single transition molecular spectroscopy that I should be aware of? It would seem that doppler broadening would be the largest contribution to the linewidth so it I may be able to fit to a simple gaussian profile instead of the more computational intense voigt. In atomic spectroscopy you usually measure a change in coherent beam intensity so the contribution from fluorescence of the atom can be made negligible in the low power unsaturated regime with polarization analyzers and other small trade-tricks. I'm not completely certain how spectrophotometers are set up with that regards, would fluorescence near the measurement frequency be something I need to take into account?


title: Falling chain with friction

text: I really need some help with a physics problem, and I guess my doubt is more conceptual, than question based. But still, let me pose it to you: > _A uniform chain of mass $M$ and length $L$ lies on a rough table, with > coefficient of friction $k$. When 1/3rd part of the chain is hanging off the > edge, it starts slipping. Find the work done by the frictional force on the > chain, at the instant the last link falls off._ Now, to find the work, I know we must use integration. The work done would be $$\int k \, p \, T \, g\, dp \tag{1}$$ Where $k$ is the coefficient of friction, $p$ is the varying length, $T$ is the mass per unit length ($= \frac{M}{l}$), and $g$ is the acceleration due to gravity. After applying appropriate limits, we get the answer. The integrand in $(1)$ simply should mean force $\times$ displacement. Now does my question come in. In the integrand, the Force quantity is defined by the variable frictional quantity, which is equal to $(k \, p \, T \, g)$, so where did the quantity of displacement of the chain go? Shouldn't there be an extra $p$ in the intgrand, such that it becomes $(k \, p^2 \, T \, g)$? Or is the notation $dp$ itself used to signify the displacement? I know the question is a bit complicated, but please help me out. It will be highly appreciated.


title: Irreducible decomposition of higher order tensors

text: I am familiar with the notion of irreps. My question refers simply to tensor representations (not tensor products of representations) and how can we decompose them into irreducible parts? For example, a rank 2 tensor is decomposed into an antisymmetric part, a traceless symmetric and its trace. What is the generalization of that for higher rank tensors? Could someone provide an example for, say rank 3 or 4? Thank you


title: Possibility of determining the mass of the water without a scale?

text: Possibility of determining the mass of the water by knowing water volume, water temperature and atmospheric pressure. I want to know if I can determine the mass of $V=0.01\,m^3$ of water in $T=298\,k$ and $P=1\,atm$. So what formula should I use?


title: Order = Energy = Mass?

text: Here is a following problem I encountered when chatting about physics with my friend: Let us imagine a classical example of ordered state of matter in thermodynamic sense: let's take a cylinder filled with air in normal pressure. Now let's make order in the chaotic movement of air particles and arrange their speed so that all the particles are coherently revolving around the axis of the cylinder in such a way, as to change only direction of the movement of each particle, not its velocity; so that the energy (and energy distribution) remains unchanged. It is clear, that immediately after such magical modification, there is a lot of energy that can be extracted from such system, for instance if we have had installed a wind turbine inside the cylinder. So the question is: **Will the system of cylinder with air inside change its weight if we change its entropy only?** The question is related to Is a hard drive heavier when it is full?, but substantially different, because there is no potential energy involved here.


title: laser spectral width vs. linewidth

text: I have been racking my brains over the differences between laser spectral width and something called the linewidth. The linewidth was written about in detail by Henry in 1982. The spectral width is the width at -20dB down from peak of the wavelength spectrum of the laser. I am looking at some laser data right now that is saying that laser x has 10 kHz linewidth and 60 pm spectral width. You can convert spectral width from frequency to wavelength as in this article and I have done that calculation. By that calculation, a laser with a linewidth of 10kHz should have a wavelength width of about 1x10^-6 nm, not 0.06 nm. Linewidth is often measured with self-heterodyne technique, not a spectrum analyzer. What am I missing?


title: How much force is required to lift the back of a 2,480 lb car in the air from the bumper?

text: I don't know the **EXACT** measurements, but I can closely guess that the bumper is about 18 inches from the ground, and the engine is in the front, yes(the engine is about 20 inches from the front bumper inside the hood, and about 23 inches centered between the hood on both sides). I don't know what the engine weighs, but the car's overall curb weight (whatever that means) is 2,480 lbs. Convert to any other weight measurements if you wish, but I'm most used to pounds. I can assume you can easily find the specs of the size ... it's a Toyota Corolla 2002 blue SE limited-edition model. Specs: It is about fourteen feet and six inches long. It is five feet and size inches wide from both doors on the front left to right seats. The wheelbase is around eight feet. The car is in total four feet and five inches from the floor to the roof in height. I don't know if this question is relevant or meets the standards here, but I would really appreciate someone decent at physics and maths to help me find the weight here that is lifted from the back bumper's height, and the total force necessary to raise both wheels in the air about one inch. Some have said that since there's shocks, the weight is favorable to the lifter somewhat, but to what degree I'm uncertain. Any help would be greatly appreciated in your findings on this issue.


title: Decay from excited state to ground state

text: People frequently speak about an atomic system decaying from an excited state to the ground state. However, both the ground states and the excited states are defined as eigenstates of the Hamiltonian operator for the system. This implies then that up to a time-dependent complex phase, they are invariant under evolution according to the Hamiltonian for the system. How can it be then that there is a decay from an excited state to a ground state? I have tried to give this an interpretation in terms of unstable equilibrium (that is, if we have a excited state, it is actually an eigenstate, but if we modify it a little bit, it becomes something will evolve to the ground state). However, I don't think this works, since the evolution under the Hamiltonian of the system will leave invariant (up to a time-dependent complex phase) the amplitudes of the state when expanded in a basis of eigenstates. My current guess is that it is necessary to consider some kind of noise to explain this, but I don't have any idea about how would the particular details work.


title: Hawking Radiation as Tunneling

text: Firstly, I'm aware that Hawking radiation can be derived in the "normal" way using the Bogoliubov transformation. However, I was intrigued by the heuristic explanation in terms of tunneling. The story goes that a virtual particle- antiparticle pair arises just outside the horizon. The negative energy partner tunnels in and the positive energy partner escapes to infinity as Hawking radiation. (Conversely the pair can be though of as arising inside the horizon and the positive energy parnter tunneling out). This reference gives a treatment of this process, using the WKB approximation to handle the tunneling. The thing that's confusing me about this, though, is the origination of the particle/antiparticle virtual pair in the first place. Taking the case for the moment where the pair is an electron/positron, I'm used to seeing such pairs arise in QED, following the QED rules. So for example an electron/positron pair might arise in an oyster diagram, contributing to the vacuum vacuum amplitude. ![enter image description here](http://i.stack.imgur.com/YT7dw.jpg) However in the tunneling description of Hawking radiation, no one seems to mention processes such as this. Only the electron and positron make an appearance. Why is there no photon involved? Is this simply because the tunneling explanation can only ever be heuristic and shouldn't be pursued too far? Or perhaps it's because Oyster diagrams belong to perturbation theory and tunneling is essentially non-perturbative and QED vertices are completely irrelevant?


title: What type of symmetry specifies in presence of what each not-charged particle (for example, neutrino) should have an antiparticle?

text: I think that charging symmetry assumes antiparticle presence, which has an opposite charge sign . And what symmetry assumes existence of an antiparticle at a neutral particle


title: Reflectance and Absorption properties of X-rays

text: Are there any materials that are mostly "transparent" to x-rays? Such as glass would be to the visible spectrum? What about materials or surfaces that reflect x-rays? Does most metal reflect x-rays? What about materials that are mostly opaque? I know lead can absorb x-rays, but is there anything else that is healthier?


title: gravity affecting a molecules speed?

text: Why are the poles colder than the rest ofthe planet?Is the gravity stronger making molecules spin slower?How does gravity affect heat and hydrogen bonds? And what affects gravity?A little off topic but Also, if you have alot of copper in your blood versus iron,does it affect your temperature?How does oxygen affect temperature?


title: How does a small radius connector affect gas flow?

text: Assume I have a 1m3 tank of air at 100 psi, they are going to release to a 1m3 tank which has air at 5 psi. There is a meter long pipe connect between two tanks. How to calculate the flow speed ratio if the pipe is 1" diameter & connection at 5psi tank is also 1" and the pipe is 1" diameter and the connection at 5 psi tank is only 0.25" ??


title: How does the radius and length of a pipe affect vacuum performance?

text: Can I use Poiseuille's equation? Also If I have a vacuum pump which says that its performance is 30 cubic feet per minutes and I have two pipes (one is radius R1 and L1, the other one is 2*R1 and 0.5*L1), will the vacuum do the same 30 CFM?


title: Are reversible adiabatic processes always isentropic?

text: If my understanding is correct, neither reversible nor adiabatic processes are necessarily isentropic. But are reversible adiabatic processes always isentropic?


title: Measure the phase of a quantum field?

text: Is it possible to measure the phase of a quantum field or quantum particle, as an observable?


title: Matrix elements of momentum operator in position representation

text: I have two related questions on the representation of the momentum operator in the position basis. The action of the momentum operator on a wave function is to derive it: $$\hat{p} \psi(x)=-i\hbar\frac{\partial\psi(x)}{\partial x}$$ **(1)** Is it ok to conclude from this that: $$\langle x | \hat{p} | x' \rangle = -i \hbar \frac{\partial \delta(x-x')}{\partial x}?$$ And what does this expression mean? **(2)** Using the equations: $$ \frac{\langle x | \hat{x}\hat{p} | x' \rangle}{x} = \frac{\langle x | \hat{p}\hat{x} | x' \rangle}{x'} = \langle x | \hat{p} | x' \rangle $$ and $$\langle x | [\hat{x},\hat{p}]|x'\rangle=i\hbar \delta(x-x')$$ one can deduce that $$\langle x | \hat{p} | x' \rangle = i \hbar \frac{\delta(x-x')}{x-x'}$$ Is this equation ok? Does it follow that $$\frac{\partial \delta(x-x')}{\partial x} = - \frac{\delta(x-x')}{x-x'}?$$


title: Gaussian surfaces and Gauss law

text: Does Gauss law holds for any closed surface or it only holds for only Gaussian surface. Are every closed surface a Gaussian surface?


title: How can I tell if a system has a quadrupole moment?

text: We know that gravitational waves are emitted (at least in GR) when the system has a time-varying quadrupole (or higher) moment. My question is _Is it possible to easily tell (e.g. just by looking) if a system has such moments?_ Is there some kind of symmetry to look for that lets you know that the system will emit such radiation?


title: Observation of light bending spacetime

text: Has radiation or energy bending spacetime ever been observed? If not, is it likely that it ever will, assuming current technology? * * * **Note:** This is not a question of _space_ bending _light_ , but of _light_ bending _space_.


title: Spontaneous motion in a wind tunnel

text: Suppose that a hollow truncated cone is placed in a wind tunnel with a steady wind speed $V$. The cone is placed in such a way that it's base of area $A_1$ faces the wind (rather than the other side of area $A_2$ for which $A_1>A_2$). Here's a contradiction: Supposing that $V_2$ is the speed for which wind appears from that end of the cone of area $A_2$, using the conservation of mass law, we get $A_1V=A_2V_2$ and hence $V_2>V$. Now using the relation $\sum F=\dot{m}\Delta V$ (which results from the more general relation $\sum F=\frac{\partial}{\partial t}\int_{C.V}V.\rho (dv)+\int_{C.S}V.\rho (VdA)$) we see that a net thrust is generated which pushes the cone forward and against the wind, but this is impossible. What's wrong with this argument? I've been giving it some thought but I can't figure out a way to solve it.


title: Limitations on the choice of axis of rotation regarding rolling wheels

text: Consider a situation where a wheel is rolling without friction on a level surface. Call the center of the wheel $C$, the point where the wheel contacts the ground $G$, and some arbitrary other point on the rim of the wheel $P$. The normal force has no torque when $G$ or $C$ is picked as the axis of rotation, but it does if $P$ is chosen for any point (minus the top of the wheel). If a physics problem then says to pick the force that gives the largest torque, how am I meant to choose if the axis of rotation choice changes the answer? The physics book I have states that $G$ or $C$ are points at which an axis of rotation may be placed because they both have the same angular acceleration around each other. This seems fine to me; I can visualize the concept of the $C$ rotating about $G$. The trouble is that it seems to be the same situation for any $P$. Working out the math, it does seem that every point on the rim chosen as the axis of rotation is such that the center of mass rotates about the chosen point at the same speed as the chosen point rotates about the center of mass. What is going on here? How does one 'get a feel' for these problems where it seems that the axis of rotation choice is so critical?


title: Decomposition of deformation into bend, stretch and twist?

text: I'm wondering is there any way to decompose the deformation of an object into different components? For example, into stretching, bending and twisting part respectively? The decomposition could be applied to any description of deformation, either to deformation gradient tensor, strain tensor or stress tensor. To my knowledge, there is some literature using the decomposition of deformation gradient tensor into rotation and deformation part. But what I want is to further decompose the deformation, leave away the rigid transformation like translating and rotating. Thanks!