title: Eigenstate of position+momentum?

text: I'm studying Quantum Mechanics on my own, so I'm bound to have alot of wrong ideas - please be forgiving! Recently, I was thinking about the quantum mechanical assertion (postulate?) that states with a definite value for an observable are eigenvectors of that observable's operator. This might sound kind of silly, but what if, for a one dimensional particle, one decided to measure position+momentum? It's easy to construct the corresponding operator (in atomic units $\hbar=1$): $x-i\frac{d}{dx}$ It's pretty easy to find an eigenfunction of this operator: $\exp(\frac{x^2}{2i}-cx)$. So, why can't the uncertainty relations be violated in such a case, if I could, say, measure the position of the object with this wave function, and immediately know the momentum (c-position)? I understand that if the wave function collapses into a delta function (as it would in an idealized position measurement) then the momentum wave function would necessarily be a constant (or at least have constant mod) over all space, meaning that the momentum would be completely unknown. UPDATE:   1. I understand that the momentum wave function is the Fourier transform of the position wave function, and that the uncertainty relations can be shown with purely mathematical arguments about the std deviaion of fourier transform pairs - I have no problem with the uncertainty principle in general.    2. I'm very appreciative for the answers, but I don't think the question was answered completely: I think there was some confusion (my fault and mostly on my part) about exactly what was meant about "measuring the quantity x+p". I was imagining that if a system's vector is an eigenvector, with eigenvalue k, of some observable O, then any measurement of O will come out to k and will leave the system in that same eigenvector state - even if O is some kind of "compound observable". So if I have a quantum system in some state that is an eigenvector of an operator that involves two quantities, I thought one could measure only one of the quantities and then infer the second one using the known eigenvalue (in much the same way that the spin of an entangled electron can be measured and then immediately the other spin will be known because the entangled state was an eigenvector of the "spin 1 + spin 2 operator"). If I had instead asked if x1 + p2 were measured for a two particle system, then the first position could be measured and immediately p2 would be known (if the system started out in the appropriate eigenvector state), right? I think the problem in my original reasoning was the lack of freedom in the momentum wave function once the position wave function collapsed - i.e. one determines the other. In a several part system one part's wave function can be manipulated independently of the others.


title: Deriving the canonical ensemble

text: I'm confused by the derivation of the canonical ensemble, namely the origin of the probability density, that is the Boltzmann factor. Here's what I have: We have a system of particles with $(N_{tot},V_{tot},E_{tot})$ in thermodynamical equilibrium that we divide into two subsystems, $A$ and $B$. Let $(N,V,E)$ describe $A$, and let $(N',V',E')$ describe $B$. We have the relations $$ N+N' = N_{tot} $$ $$V+V' = V_{tot}$$ $$E+E' \approx E_{tot}$$ where in the last one we've neglected any potential interaction between the particles, arguing that the number of degrees of freedom in any interaction is orders of magnitude smaller than the total amount of particles, because interactions are short ranged. Now, consider the probability that subsystem $A$ has energy $E$. This is the ratio of the number of microstates of the entire system wherein $A$ has energy $E$ to the number of microstates such that the entire system has $E_{tot}$: $$P(A \mbox{ has }E)= \frac{\Omega_{tot}(A \mbox{ has }E)}{\Omega_{tot}(E_{tot})}$$ but since we've neglected the interactions, we can factor the numerator: $$P(A \mbox{ has }E)= \frac{\Omega(E)\Omega ' (E')}{\Omega_{tot}(E_{tot})}$$ where the omegas follow the same convention with primes and subscripts as the other variables. But now I don't understand how to get the Boltzmann factor... this relies on having $$P(A \mbox{ has }E) = C \cdot \Omega ' (E')$$ But I don't see why that's true... $\Omega(E)$ isn't constant, is it?


title: How to calculate gravitational force

text: Suppose the centre of mass of 2 bodies coincide , then how will we calculate the gravitational force between the 2 bodies ??


title: What is the difference between the KDEMO and DEMO projects?

text: I'm writing a paper on the future of fusion technology and I can't seem to find the difference between Europe's DEMO experiment and Korea's KDEMO except for the fact that they are both planned to follow ITER which involves Korea and many European states and that they of course will be in different locations.


title: Integrals of Motion for s Degrees of Freedom

text: From Landau & Lifshitz, _Classical Mechanics,_ the number of integrals of independent integrals of motion for a system of $s$ degrees of freedom is $2s-1$. I am considering a spherical pendulum in absence of gravity. For this problem $s=2$, so I expect 3 independent integrals of motion. The first two are the angular momentum in the $\theta$ and $\phi$ direction. The energy is also a constant of motion, but in the absence of motion it is a function of the angular momentum since $$L=T=\frac{m\ell^2}{2}(\dot{\theta}^2+\dot{\phi}^2\sin^2(\theta)).$$ I have 3 integrals of motion but 2 are independent. Did I miss one, or have I misunderstood the statement from Landau? The linear momentum should not be conserved, so should I include integrals of motion of ignorable coordinates?


title: Does induced current affect the permeability of a material?

text: For electromagnetic wave propagation, we can typically characterize a medium by conductivity $\sigma$, permeability $\mu$, and permittivity $\epsilon$. Consider if this material had current passing through it. Would it possibly change the permeability $\mu$? So if you had a rod of mu-metal for example, you know the permeability $\mu$ but does that change if you induce current through the rod? Update: So I understand that B-H the relationship has a point where the material saturates, and increasing the incident field only causes a small increase in the induced field, and effectively reduces permeability $\mu$. However, in the above example, assume that the incident H field is small enough that it does not reach saturation. What I am talking about is if you connected wires at the 2 ends of the rod and connected it to a current source (AC or DC), does that change the permeability $\mu$? Perhaps you guys already understood this but let me know if your answer changes.


title: Confusion between the de Broglie wavelength of a particle and wave packets

text: So I learned that the de Broglie wavelength of a particle, $\lambda = \frac{h}{p}$, where h is Planck's constant and p is the momentum of the particle. I also learned that a quantum mechanics description of a particle is a wave packet. I learned that a wave packet is a summation of different basis functions overlapped over each other at say $x = 0$, and these basis functions are the wave functions, $\Psi(x,t)$. Or is it the probability density, $|\Psi(x,t)|^2$ ??? Please correct me on this. I learned that the more localized the wave packet is in position space, the more un-localized or uncertain you are about the spread of momentum functions. Please also edit my statement I just said because I don't think I stated it in the best way. My question is, you have a particle that is represented by a wave packet that is localized, thus, it has a spread of momentum, so how can you then know it's de Broglie wavelength? Do you average together all the different momentum the particle has and then plug that average momentum into $\lambda = \frac{h}{\langle p\rangle}$ ??


title: Size of the universe

text: This is really a follow up to the Shape of the universe question. In the first answer to the question, Ted Bunn says: > However, the best available data seem to indicate that the Universe is very > nearly spatially flat. This means that, if we do live in a 3-sphere > Universe, the radius of the sphere is very large, and the distance you'd > have to travel is much larger than the size of the observable Universe. My question is, can the available data be used to set a lower bound to the size of the Universe? In other words, suppose the Universe was actually shaped like a 3 sphere. Then based on our best data, what would the "radius" of that sphere be? How much larger is that than the observable universe?


title: Mathematically rather than physically speaking, is there something "special" about 10 (or 11) dimensions?

text: As I understand it, string theory (incorporating bosons and fermions) "works" in $9+1=10$ spacetime dimensions. In the context of dual resonance theory, I've read descriptions of why that is "physically", i.e., how in the process of formulating a quantum theory of relativistic interacting (closed and open) strings, the only way one can do things like get rid of ghosts, for example, is to have ten dimensions. I'm looking for a way to appreciate more intuitively this restriction on dimensionality. Is there something special about ten-dimensional spaces purely from the point of view of mathematics? Without reference to the particular detailed exigencies of building a physical theory based on strings, does ten- dimensional topology have some unique features that are lost when one dimension is added or subtracted? Another way to pose the question: For purposes of this discussion, let's stipulate by fiat that the universe has $8+1=9$ dimensions. Would we humans simply be back to the drawing board in terms of coming up with a unified theory of the fundamental forces? Would anything in string theory be salvagable? I realize that M-theory adds a dimension. So if someone wants to answer the question by telling me what's unique or special about eleven dimensions, that's fine.


title: Apparatus used to test violations of Bell's inequalities

text: I believe most people are familiar with Bell's Theorem. If I understand correctly, the violations of Bell's inequalities are possible due to the existance of entangled quantum states. Although popular discussion often circles around nonlocality questions and doubts about what the experiments say with regards to local realism, we rarely see lengthy discussions about the implications of creating the apparatus itself. I wonder if it is possible to reasonably view the apparatus as a sort of quantum state detector? In asking this, I am not thinking about a detector as a device that detects a particle, but as a device that has detected the existance of a possible state of particles. If one thinks of the spin correlation diagram in the article, one might interpret the possible outcomes as two possible system response curves (or possibly zero or one). It seems that if the apparatus is properly tuned to a particular type of particle system, one should always see the quantum result. Is there some clarity that can be brought to this issue?


title: What is the general conditions for non-zero polarization?

text: I only know that for a system to have non-zero polarization it should break inversion symmetry. What about others requirements?


title: What is $l_i$ in The wetted perimeter

text: The wetted perimeter can be defined mathematically as $$P=\sum_{i=0}^{\infty} l_i$$ what is this means that $l_i$ is the length of each surface in contact with the aqueous body? the length of each surface of what is in contact with aqueous body?


title: Free energy a continuous function of temperature but may not be differentiable everywhere?

text: So according to my understanding, the free energy of the system should be a continuous function of temperature. This is because if the free energy is not continuous at temperature T, then at this temperature the free energy can have two values (i.e. two states) and the two states will mix to eliminate the difference in their free energies. Is that a reasonable explanation for the continuity of free energy? The free energy is a continuous function of temperature does not mean it is differentiable everywhere, because at the point where the phase coexistence start to occur the free energy is not differentiable. Is that the case?


title: Nature of Fields in QFT

text: I'm not exactly an expert in quantum physics, but this seems to be a simple question, and I can't find an answer anywhere! There are specific types of fields used in physics: scalar fields (i.e. as in the case of the Higgs boson), vector fields (i.e. as in magnetic fields), tensor fields (i.e. as in general relativity), etc. But what types of fields are used in QFT to model elementary particles? Is my confusion simply a result of me thinking in purely classical terms?


title: Do electric and magnetic lines of force physically exist?

text: As per my imagination any thing can't impose force on the other by not giving even a touch(i,e action at a distance). So I thought there must be some physical existence of lines of force. Although virtual particles hypothesis has been proposed. Up to my knowledge virtual particles are just said to be disturbance (field) and not the one what we thought as materialistic "particle". Anyway I have heard about Faraday speaking about such similar view about there physical existence. But I do have a naive idea about it. I would greatly appreciate if any one can input something about your view on the physical existence of force lines. It would be more appreciable if any one can speak about Faraday's and Maxwell's approach on this concept. ![enter image description here](http://i.stack.imgur.com/tHnpw.png) **I am providing Faraday's and Maxwell's basic views about the concept, so that anyone can speak about it.** **FARADAY'S VIEWS:** At first Faraday considered the physical reality of the lines of force as a possibility, yet several scholars agree that for Faraday their physical reality became a conviction. One scholar dates this change in the year 1838. Another scholar dates this final strengthening of his belief in 1852. Faraday experimentally studied lines of magnetic force and lines of electrostatic force, showing them not to fit action at a distance models. In 1852 Faraday wrote the paper "On the Physical Character of the Lines of Magnetic Force" which examined gravity, radiation, and electricity, and their possible relationships with the transmission medium, transmission propagation, and the receiving entity. **MAXWELL'S VIEW:** Initially, Maxwell took an agnostic approach in his mathematization of Faraday's theories. This is seen in Maxwell's 1855 and 1856 papers: "On Faraday's Lines of Force" and "On Faraday's Electrotontic State". In the 1864 paper "A Dynamical Theory of the Electromagnetic Field" Maxwell gives scientific priority of the electromagnetic theory of light to Faraday and his 1846 paper "Thoughts on Ray Vibrations". Maxwell wrote: Faraday discovered that when a plane polarized ray traverses a transparent diamagnetic medium in the direction of the lines of magnetic force produced by magnets or currents in the neighborhood, the plane of polarization is caused to rotate. The conception of the propagation of transverse magnetic disturbances to the exclusion of normal ones is distinctly set forth by Professor Faraday in his "Thoughts on Ray Vibrations." The electromagnetic theory of light, as proposed by him, is the same in substance as that which I have begun to develop in this paper, except that in 1846 there were no data to calculate the velocity of propagation. **I hope anyone from quantum physics can speak about virtual particles.**   **Here is a opinion ofMatt Strassler (click on this) about Virtual particles hypothesis**


title: Why water wont leak out in this case? (Picture provided)

text: just curious why the water in the aquarium tank would not leak out as shown in the picture provided below? The question is in the red circle. The water level in the aquarium tank is definitely higher than the hand entrance from what it seems. How they did that? What sorcery is this? ![enter image description here](http://i.stack.imgur.com/ENvlI.png)


title: What causes atoms to have their specific colors?

text: I understand that light (color) is part of the electromagnetic spectrum, and that it depends on what wavelengths are reflected/absorbed. Though what property of an individual atom gives it its color? Do electrons, neutrons, and other subatomic particles have color? **Basically: What intrinsic property causes the differences between how the varying wavelengths of light are reflected at the atomic scale?** Also, how do photons factor into this?


title: Is there a well known theory that states all particles are made up of smaller particles and so on and so forth?

text: I've contemplated, for some time, the possibility that all things are made up of smaller particles, and all smaller particles are made up of even smaller ones. I'm wondering if there are any popular theories that specifically support or deny this possibility.


title: An Introduction to particle acceleration and particle accelerator

text: I am preparing to write my paper about particle acceleration recently, and i need some general knowledge to get my feet wet about it.   May you please give me an introduction of what particle acceleration is, the concept, the basic mechanism of particle accelerator, the general principle, or please refer me a link of particle acceleration. Thanks! Regards.


title: Turned to steel in the great magnetic field

text: This is obviously a "fun" question, but I'm sure it still has valid physics in it, so bear with me. How great of a magnetic field would you need to transmute other elements into iron/nickel, if that's in fact possible? The magnetic field near a neutron star, roughly of order 10^10 tesla, is so strong that it distorts atomic orbitals into thin "cigar shapes". (The cyclotron energy becomes greater than the Coulomb energy.) Certainly if a solid crystal were placed in such a field it would become very anisotropic, and at some field strength the lattice constant in the direction transverse to the field could become small enough for nuclear fusion rates between the nuclei to become non-negligible. How high do we need to crank up the field before the nuclei all equilibrate to the absolute energy minimum of iron and nickel in, say, a matter of hours or days? Update: From http://dx.doi.org/10.1086/152986 it appears that matter in strong magnetic fields forms into strongly-bound 1D chains along the field lines, which are only weakly bound to each other, and the parallel and transverse lattice constants are actually comparable.


title: Low pressure, pulsating, series of water droplets

text: I want to create a stream of water that emits only a droplet of water, waits a few milliseconds, and then continues. The important thing is that I need to create a visible gap between drops. Considering the desire to have droplets created with a consistent size and shape (asked in this question), how would one go about creating a stream analogous to a morose-code-stream of droplets? **[Edit]** I'm considering having a solenoid valve connected to a tube pointed downward, where capillary action holds the water in place. The solenoid releases the amount equal to one droplet of water. What I'm having trouble with is making the drops look semi-uniform as they fall. Here is an artistic rendering of what I'm trying to accomplish: ![alt text](http://i.stack.imgur.com/2uYXm.jpg)


title: Frequency of a tuning fork in a vacuum

text: Consider this equation of a damped harmonic oscillator such that: $$ \ddot{x}+2\gamma\dot{x}+\omega^2_0=0 $$ with: $\gamma=\frac{b}{2m}$ and $\omega_0=\sqrt{\frac{k}{m}}$ Finally, we know that the equation x(t) should be of this form: $$ x(t)=e^{-\gamma t}[Acos(\omega_1t)+Bsin(\omega_1t)] $$ It is observed that the amplitude of oscillation of a tuning fork of frequency 400Hz is damped in the air by 10% in 12s. What would be the frequency of the tuning fork in a vacuum (void)? I'm really struggling to find a starting point… I've found that: $$ \omega_1=\frac{\sqrt{4mk-b^2}}{2m} $$ But I don't see where to begin to find the frequency in the vacuum. Could someone explain me how to start? Thank you in advance.


title: Why are Euler's equations of motion coupled? Physical explanation

text: I have a problem with one of my study questions for an oral exam: > _Euler’s equation of motion around the $z$ axis in two dimensions is > $I_z\dot{\omega}_z = M_z$, whereas it in three dimensions is > $I_z\dot{\omega}_z =-(I_y-I_x)\omega_x\omega_y+M_z$, assuming that the $xyz$ > coordinate systems is aligned with the principal axis. Why does Euler’s > equation of motion for axis $z$ contain the rotational velocities for axes > $x$ and $y$?_ How can one explain this physically? I mean I can derive Euler's equation of motion, but how can I illustrate that the angular velocities are changing in 3 dimensions?


title: What are the reasons for leaving the dissipative energy term out of the Hamiltonian when writing the Lyapunov function?

text: I have a problem with one of my study questions for an oral exam: > _The Hamiltonian of a nonlinear mechanical system, i.e. the sum of the > kinetic and potential energies, is often used as a Lyapunov function for > controlling the position and velocity of the system. Consider a damped > single degree-of-freedom system, $m\ddot{x}+c\dot{x}+kx=0$, where $m$ is the > mass, $c$ is the velocity-proportional damping and $k$ is the stiffness. A > candidate Lyapunov function is the Hamiltonian > $V=\frac{1}{2}m\dot{x}^2+\frac{1}{2}kx^2$. What are the reasons for leaving > out the dissipative energy term when writing the Lyapunov function?_ The only thing what comes into my mind for this question is, that a dissipative energy term in the Lyapunov function would have a "-" sign and the Lyapunov function would thus not be positive definite anymore. Is that correct?


title: Adiabatic expansion

text: I'll start off by saying this is homework, but I ask because I don't understand how the math should work (I don't just want an answer, I'd like an explanation if possible). I understand if this is still against the rules, and I apologize if it is. I have to calculate the work done by a sample of dry air on the environment in an adiabatic expansion to five times the original volume and an initial temperature of 20˚ C. The answer rounds to 100,000 (it's 99,???). So far, I have that $c_v dT = -p d\alpha$. I tried integrating, which gives $c_v T = R_d T_0 \ln{5}$ (because the limits of integration give 5 inside the natural log and $R * T_0$ is constant; the IGL gives $p\alpha = RT$). Plugging numbers in gives $T = \frac{(-287)(293)\ln{5}}{717} = -188.76$. Since the change in internal energy is equal to the negative work done (because $\delta q = 0$ in an adiabatic process), and $dT = 105$, then $\delta w = 105 * 717 = 75,285$. This answer is incorrect, but I'm not sure where I went wrong.


title: Why is $\int (dp/2\pi) |p \rangle\langle p| = 1 $?

text: In quantum mechanics, why is $\int (dp/2\pi) |p \rangle\langle p| = 1 $ where $|p \rangle$ represents momentum eigenstate?


title: How to interpret a negative failure rate?

text: In statistical engineering the "hazard rate" of a distribution is defined as: $$r(x)=\frac{f(x)}{1-F(x)}$$ where $f(x)$ and $F(x)$ are the PDF and CDF. Basically $r(x)$ is the odds that, having reached a certain point on the abscissa (usually time), you won't get any further. In the study of mechanical failure, the relevant distributions are those with an $r(x)$ that is everywhere increasing for positive x, like the Weibull distribution. My question is how would you interpret an $r(x)$ whose absolute value is everywhere increasing for positive x, but which is negative? Does the sign matter?


title: What is the purpose of sensors and pumps with less than 1 ATM pressure?

text: This question is not directly a physics questions, though related a little. Previously for sensors, and currently for air pumps, I am doing some research. I have found a digital pressure sensor on Farnell that has "maximum" 0.10 bar measurement. (I cannot find the link right now). Another is an air pump. http://www.alibaba.com/product- detail/JQT-3000-C-electric-high-capacity_1677414765.html?s=p (This is not an advertisement in any way.) This product says its maximum pressure is 35 kpa that is 0.35 bar. According to Google 1 atm = 1.01325 bar. If the maximum pressure of these devices are less than the pressure of atmosphere, then what is the purpose of them? How can a pump with less than 1 ATM pressure can do any compression? Or where am I going to be using a 0.1 bar sensor? In a vacuum?


title: Linear charge density, surface charge density and volume charge density

text: What is the difference among:   1. linear charge density,    2. surface charge density, and    3. volume charge density?


title: Does the coefficient of restitution matter in a collision?

text: Suppose two particles are colliding. Let initial velocities be $u_1$ and $u_2$ .Their final velocities are $v_1$ and $v_2$. Now $u_1$ and $u_2$ are known. By intuition I can understand that $v_1$ and $v_2$ will depend on the coefficient of restituition and hence on their material. Now I have an equation from conservation of linear momentum and another for the coefficient of restituition. I can now find $v_1$ and $v_2$. So it looks as if $v_1$ and $v_2$ depend upon the coefficient only. But in my physics textbook I have a solved problem. There, a ball hits a wall and initial and final velocities and the mass of the ball were given. They have basically calculated the average force and impulse. They are saying that the impulse remains same and the maximum force along with the time of collision will depend on the softness or hardness of the ball. But that means we have the same $v_1$ and $v_2$ for different coefficients of restituition. So where is my thinking wrong?


title: Can anyone estimate what proportion of water remains after I flush a toilet?

text: Imagine I have a clean toilet with some amount of water in the bowl. When I flush the toilet much of that water will be displaced by the tank's water. I want to work out (or model really) what amount of the original toilet bowl water will remain after the flush. If there was B liters of water in the bowl and T liters was released from the tank I'd expect the proportion to be at most B/(B+T) of original water to tank water.. but this doesn't take into account how the water is released. I assume this is studied in the design of toilets, so links to existing research would be interesting too! I also appreciate that it will vary for each toilet design, so feel free to answer the question for any common type or model; I'm most interested in a way of estimating or modelling it.


title: Does a magnetic field induce an emf in a loop of wire

text: Is it true that any magnetic field induces an emf in a loop of wire? If It it is true then kindly explain me with an example


title: is cosmic expansion related with IR divergencies?

text: This question is related to renormalization, but in the IR limit. It is assumed that unitarity does take care of IR divergencies in interacting theories like QED. But how would one interpret physically a theory that would effectively diverge at large wavelengths? Do we expect De-Sitter cosmologies to be free of IR divergencies? **why?**


title: String boundary conditions

text: I'm reading Polchinski and am confused about equation (1.3.13), $$\gamma_{\tau\sigma}\partial_\tau X^\mu-\gamma_{\tau\tau}\partial_\sigma X^\mu=0~~~~~\text{at}~~~~~\sigma=0,l.$$ It says that this comes from $$\partial^\sigma X^\mu(\tau,0)=\partial^\sigma X^\mu(\tau,l)=0$$ along with the gauge conditions $$\partial_\sigma\gamma_{\sigma\sigma}=0;~~~~~\det\gamma_{ab}=-1,$$ but I really don't see it. How does the metric come into play?


title: The ${\cal N} = 3$ Chern-Simons matter lagrangian

text: This question is sort of a continuation of this previous question of mine. I would like to know of some further details about the Lagrangian discussed in this paper in equation 2.8 (page 7) and in Appendix A on page $31$.   * On page 7 the authors introduce the idea of having a pair $(Q, \tilde{Q})$ for the matter and that these are ${\cal N}= 2$ ``chiral multiplets" transforming in adjoint representations of the gauge group. But on page $8$ they seem to refer to the same matter content as being $N_f$ hypermultiplets.  What is the relation between these two ways of thinking about it? I haven't seen a definition of a "chiral multiplet" and a "hypermultiplet" for $2+1$ dimensions.   * If the gauge group is $U(n)$ and we are working in a representation $R$ of it then should I be thinking of $Q$ and $\tilde{Q}$ as matrices with two indices $i$ and $a$, $Q_{ia}, \tilde{Q}_{ia}$, such that $1 \leq i \leq N_f$ and $ 1 \leq a \leq dim(R)$? And their transformations are like (for a matrix $U$ in the $R$ representation of $U(n)$), $Q'_{ia} = U_{ab} Q_{ib}$ and $\tilde{Q}'_{ia} = U^*_{ab}\tilde{Q}_{ib} = \tilde{Q}_{ib}U^{\dagger}_{ba}$ Is the above right?   * In appendix $A$ (page 31) what is the explicit form of the ``fundamental representation of $USp(2N_f)$" that is being referred to? Is that the matrix $T$ as used in $A.3$ on page 31?    * In equation $A.3$ I guess the notation $()$ means symmetrization as in,  $s^{m}_{ab} = \frac{4\pi}{k} q_{A(a} T^m q^A_{b)} = \frac{4\pi}{k} (q_{Aa}T^mq^A_b + q_{Ab}T^m q^A_a)$ I guess it is similarly so for $\chi ^m_{ab}$ ?   * In equation 4.4 (page 31) is the first factor of $\frac{k}{4\pi}CS(A)$ equal to equation 2.4 on page 5?   * In the same expression for $A.4$, what is the meaning of the quantities     * $D^{ab}$ as in $Tr[D^{ab}s_{ab}]$?      * $\chi _{ab}$ (..what is the explicit expression which is represented by $Tr[\chi ^{ab} \chi _{ab}]$ ?      * $\chi$ as in $Tr[\chi \chi]$ and the last term $iq_{Aa}\chi \psi ^{Ab}$ ? (..is this $\chi$ the fermionic component of the gauge superfield and different from $\chi ^{ab}$?..) This Lagrangian is notationally quite unobvious and it would be great if someone can help decipher this.


title: regarding nuclear magnetic resonance

text: In NMR experiment we provide an external radio frequency signal which will induce transition of nuclear spins to higher states,and then it returns to ground state precessing around the magnetic field in larmor frequency,which is the same as that of absorbed radio frequency.One thing i don't understand is the underlying reason for resonance. Why is it that a system absorbs a certain frequency called the natural frequency? Another thing is both protons and neutrons have different magnetic moments so they should have different larmor frequencies(right?),but we obtain only one resonance frequency in the experiment so which one does it correspond to?Another thing is how do neutrons which are neutral particles have magnetic moment?


title: Internal Energy Terminology

text: I am interested in finding out whether the term 'Internal Energy' can be used in both of the following cases:   * In the case of a macroscopic body upon which light is incident, the internal energy would designate the total energy absorbed by the body, causing it to heat up (in the absence of other heating mechanisms)   * In the case of a low-density system of atoms upon which light is incident and absorbed, the internal energy would designate the total amount of energy absorbed by the atoms and therefore would relate to the sum of the excitation and kinetic energies of the atoms. Are both uses of the term 'Internal Energy' correct, or would it only apply to one of the cases? In both cases it would refer to the total energy absorbed by the system from radiation, but how the energy is distributed in the system (kinetic vs excitation) would depend on the type of system.


To start or change the movement, we need a force. This force, according to Newton's second law, is variation of quantity of motion from the object per unit of time.Therefore, if the variation does not exist, the amount of movement of objects is conserved.This conclusion allows enunciate the Newton's first law as follows: The amount of movement of an object is preserved if the resultant of the forces that act on it is null. So, Does the movement of things in the universe will cease one day?


title: When can I use equations of motion?

text: The three equations are: $$v = u+ at , $$$$s = ut + \dfrac{1}{2} at^2 , $$$$ v^2 = u^2 + 2as.$$ Can I use them for both finding instantaneous and average quantities? Suppose I want to find the final instantaneous velocity then can I use the first equation or if I want to find average velocity over a time period, can I use the 1st equation? Plz explain me elaborately.


title: On-site repulsion and Pauli exclusion

text: Been studying hopping conduction and something that everyone is taking for granted is bothering me. Let's say we have a bunch of sites that are either unoccupied, singly occupied, or doubly occupied. Due to on-site Coulomb repulsion the two electron levels are separated by U energy at a doubly occupied site. Now everyone is saying that the two electrons on the double site are in the spin singlet state due to, I assume, Pauli exclusion. However the two electrons are not in the same energy level - they are separated by U so why is there a restriction on their spins?


title: Would a sneeze by a cosmonaut in a spacesuit affect his movement?

text: Naive question; feel free to shoot me down It is a truism that any motion in space would continue indefinitely unless it is opposed by an external force. If a cosmonaut were to sneeze within his/her spacesuit, would it have any impact upon their movement? I assume the suit and cosmonaut would, between them, totally absorb the force exerted by the 100mph sneeze air velocity. This would leave the cosmonaut unaffected ... probably.


title: What is the mechanical explanation for obstruction of motion by an object?

text: A large box (1) is on a flat surface and is towed by a constant force in the horizontal direction, left to right, but it is not moving due to the presence of a little box (2) in front of the larger one. The force that box (2) exerts on the larger box (1) is the same force that the large box (1) exerts on the little box (2), $F_{12} = F_{21}$, right? Is this why the larger box is not moving?


title: How do I relate the direction of a compass needle to the direction of a current?

text: >   1. A compass needle is placed above the wire points with its N-pole toward > the east. In what direction is the current flowing? >   2. If a compass is put underneath the wire, in which direction will the > needle point? > I have quite a few problems on this for my Physics homework tonight, and I am utterly confused. The book says nothing about direction besides stating the right hand rule. What confuses me is that the compass needle points with its N-pole towards the east...so would that change the direction of the current? and then also, what is the difference between the magnetic field above versus below the wire?


title: Why is the Fraunhoffer Pattern the Fourier transform of the slit?

text: Why is it that the fourier tranform of the slit (with value 1 where it's open?) gives the fraunhoffer diffraction pattern? Why are these two paired?


title: What prediction led to the vacuum catastrophe?

text: The disagreement between predicted and measured energy density of the vacuum is one of the great unsolved problems of science. According to this book the predicted energy density was obtained as follows: > ...we can adopt the Planck energy as a cutoff. This is the energy at which > the gravitational force becomes as strong as the other three fundamental > forces of nature (i.e. the scale at which we expect the current theory to > break down). This energy is about $10^{19}GeV$. My questions are:   1. Why say that gravity becomes as strong as the other forces at the Planck scale? My understanding is that the Planck scale is characterised by our not being able to make meaningful statements about the laws of physics any more. Isn't this a bit like taking conditions at a singularity and applying them everywhere?   2. Do the magnitudes of all forces approach equality as we approach $10^{19}GeV$? Electromagnetism and the weak force unify at $100GeV$, where does $10^{19}GeV$ come from?   3. Come to think of it, what does $10^{19}GeV$ have to do with the Planck scale at all?   4. Why use a cutoff? Shouldn't the theory apply at above-Planck scales? I only have an armchair understanding of physics at this level, but the approach described above doesn't pass the smell test. I'd appreciate a better explanation.


title: Who popularized $E=mc^2$?

text: When asking a layman for a mathematical or physical equation you will almost certainly get the answer "$E$ equals $m$ $c$ squared". In fact, $E=mc^2$ is something like a symbol of physics in popular culture. I'm wondering which historic events lead to the enormous popularization of this equation. There are many other important and deceptively simple equations in physics and math such as $e^{\pi i}=-1$, $E_\text{pot}=mgh$, $\nabla \cdot B=0$ and $H\psi=E\psi$, so why exactly did this one equation become so popular while the others are barely known to the public? In particular, I would like to know _when_ this equation became so popular. Was everybody already talking about $E=mc^2$ in the 1900s and 1910s, or did it take some time until this equation became the representative of Einstein's theories in the public?


title: Is there an inconsistency between Quantum and Classic in probability density of harmonic oscillator ground state?

text: Consider probability densities for a particle in the lowest energy state of a simple harmonic oscillator. The quantum mechanical probability density peaks near the equilibrium point and extends beyond the sharp limits of motion predicted by classical physics. The classical probability density is inversely proportional to the classical velocity and is greatest at the endpoints of the motion, where the velocity vanishes. My question clearly is > _Is there a inconsistency between Quantum and Classic in probability density > of harmonic oscillator in it's ground state? Is thecorrespondence principle > valid here?_ ![Sec. 5.3 Eisberg Resnick Quantum Physics ](http://i.stack.imgur.com/P7Jbt.png)


title: What is chirality?

text: I actually wanted to make the title as "What is the difference between chirality and helicity"? But I didn't do that because I don't understand properly what chirality is. I have gone through this Wikipedia article: chirality to get the meaning of chirality? and what I get from there is that something is said to have chirality if it is not identical to its mirror image. But I have often seen people saying a massless particle have same helicity (handedness, I think) and chirality? Now if the chirality has the above defination then how do people say it? Again for massive particle we can have changed helicity by changing our reference frame. Ok I understand that, but How does it have anything to compare with chirality? I have read these: Helicity and Chirality and many others in this site. But didn't find(or maybe didn't understand) the answer there which I am looking for.


title: Path Integral on a circle (calculation of phase and linear independance)

text: I am reading Schulman's "Techniques and applications of path integration" chapter on Path integrals on multiply-connected spaces. In the first section he calculates the path integral of a free particle on a ring. The total path integral can be divided into equivalence classes, each containing paths with the same winding number n, i.e. no. of times we cross a fixes point on the circle. $$K(\phi^{\prime \prime}, t^{\prime \prime};\phi^{\prime}, t^{\prime})=\sum_n A_n \sum_{\phi \in g_n} e^{iS(\phi)}=\sum_{n={-\infty}}^{n=\infty} A_n K_n$$ Here $g_n$ is the set (equivalence class) of all paths with winding number n. There is a factor $A_n$ in general, which is obtained if we make the assumption that each term in the sum over winding numbers satisfies the Schrodinger wave eqn locally. Later, it can be proved that $A_{n+1}=e^{i\delta}A_n$ My questions are:   1. How do I intuitively or rigorously prove that $K_n's$ are linearly independent? Are they also orthogonal?   2. Schulman says 'magnitude of $A_0$ is fixed by unitarity to 1'. I cannot see this. I am guessing unitarity means $\int |K(\phi^{\prime \prime}, t^{\prime \prime};\phi^{\prime}, t^{\prime})|^2 d\phi^{\prime \prime}d\phi^{\prime}=1 $, but then there are a lot of cross terms and each of the integrals $\int|K_n|^2$ should be one, so that I am getting the integral diverges. How do I show $A_0=1$?


title: Why could you not travel back in the time before a tipler cylinder is created?

text: Is it because the lightcones are not tipped all the way over in which the time axis points downwards?


title: Anyone know of a piezoelectric polymer that reacts to light or electron bombardment?

text: In other words, a very small amount of light would make it rapidly expand 1/2 mm or more-- (1/1000th candlepower)--or the amount of electrons that a standard (non-lcd) TV monitor shoots at one pixel? Might there be any substance that does behave this way, maybe an organic thing? By rapidly, I mean in less than 10 seconds, ideally virtually instantly. If you don't know of one, doy you know of a person who might or otgher information resource? This is for a school project, to make a computer screen "feelable" for a blind person. Thanks much.


title: cosmic radiation muons as a source for muon-catalyzed fusion

text: taking rough numbers from here and here, it seems that with natural cosmic radiation sources, we could sustain $10^6$ muon-catalyzed fusion reactions per square-meter per minute. This would be $14 \times 10^6$ MeV of neutron energy (for D-T reactions, assuming 14 MeV per event) per square-meter per minute. In Watts-hour this would be $4 \times 10^{-12}$ KWh I think this is the right estimate, but wanted to check if someone spots a problem with my estimate cheers


title: Expansion of the Universe regarding Time and lightspeed

text: According to our current understanding, the universe is expanding... it is often said that every galaxy is moving away from us (I'm assuming that isn't true for galaxies in our local group), and it's also often said that the _farther_ a galaxy is away from us -- the _faster_ it moves away from us. We also know that light has a finite speed, so when we look at those far away galaxies, we see them as they were a long, long time ago. But (1) if the _oldest_ galaxies are moving away _faster_ , and the _younger_ galaxies are moving away _slower_ , then wouldn't that mean that the expansion of the universe is slowing down? But (2) when I'm thinking of an expanding 2d sheet of paper with dots (scaled up in both directions), then it makes sense that the farther galaxies would move faster away from our perspective.. (although if the universe were expanding at a constant rate, we would still see the farther galaxies moving faster from our perspective, but I'm assuming that it has been accounted for, and that the galaxies move even faster than that). So... why is my first "but" incorrect? Something tells me that I'm looking at it at the wrong way, but I can't figure it out xD Thanks!


title: Where can I find the measured magnetic moments of the quarks?

text: I read that the magnetic moment of the proton can be expressed as the sum of the magnetic moments of the three quarks like $\mu_p = \frac{4}{3}\mu_u - \frac{1}{3}\mu_d$. But I couldn't find a table listing the measured magnetic moments of the quarks. Can you point me to a reference?


title: Global Properties of Spacetime Manifolds

text: When solving the Einstein field equations, $$R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R = 8\pi GT_{\mu\nu}$$ for a particular stress-energy tensor, we obtain the metric of the spacetime manifold, $g_{\mu\nu}$ which endows the manifold with some geometric structure. However, **how can we deduce global properties of a spacetime manifold with the limited knowledge we usually have (i.e. simply the metric)?** For example, how may we deduce:   * **Whether the manifold is closed or exact**   * **Homology and de Rham cohomology**   * **Compactness** I know if we can establish compactness, one can easily arrive at the Euler characteristic, and hence the genus of the manifold, using the Gauss-Bonnet- Chern theorem, $$\int_M \mathrm{Pf}[\mathcal{R}]\sim \chi(M)$$ In addition, the Chern classes of the tangent bundle computed using the metric give some information regarding the cohomology. In addition, this question is really not limited to spacetime manifolds. There are many scenarios in physics wherein we may only know limited information up to the metric, e.g. moduli spaces. It would be interesting to see how one can deduce global properties. * * * This question is inspired by brief discussions on the Physics S.E. with user Robin Ekman, and I would like to thank Danu for placing a bounty; a pleasant surprise! * * * Resources, especially journal papers, which focus on addressing global properties of spacetimes (or more exotic spacetimes, e.g. orbifolds) are appreciated.


title: What causes the 'tears' in yoghurt drinks?

text: Fluid dynamical instabilities are present in many different everyday things. The famous tears in wine for example are a classical example of a Marangoni effect, where surface tension gradients due to evaporation cause an instability. I recently came across an instability that occurs when you have just emptied a glass with yoghurt drink and have only a film of the drink left on the walls of the glass. A tear-like pattern emerges over the time of about a minute that looks like shown below: ![enter image description here](http://i.stack.imgur.com/KY0lG.jpg) I have been thinking what can cause this effect. It seems somewhat similar to a Marangoni effect, but given the ingredients of the yoghurt drink I think we can safely rule out evaporation on this timescale. Another potential effect explaining this would be viscous fingering, but for that to happen I believe the gas phase has to be driven, which it isn't here. Or perhaps it is a 'surface-version' of a Rayleigh-Taylor instability? Or maybe it is related to the non-newtonian nature of the yoghurt drink? To summarize: I don't know which instability I am looking at and I would love to know what causes this pattern on the glass with yoghurt drink! Does anyone know what is causing it?


title: The feasibility of a satellite orbiting at a fixed time

text: I was speaking with some friends of mine, one of whom was an aerospace engineer. He posited the infeasibility of a hypothetical "Margaritaville Satellite" that orbited earth in such a way that wherever it was above, it was six-o'clock (the idea being, the Margaritaville Satellite is in Happy Hour 24/7), matching Earth's rotation exactly. However, there was disagreement on whether or not the satellite could actually be put in such an orbit, as if it were rotating at the exact speed of earth, it would effectively just be "falling" straight towards Earth. Is it possible to sync a satellite's orbiting with the time of day of the earth without some sort of propulsion to keep it aloft?


title: When can I use Wick's theorem?

text: Wick's theorem means that for fermions, a four point correlation function (for example) can be written in terms of two point correlation functions: \begin{equation} \langle b_l^\dagger b_l b_m^\dagger b_m \rangle = \langle b_l^\dagger b_l \rangle \langle b_m^\dagger b_m \rangle - \langle b_l^\dagger b_m^\dagger \rangle \langle b_l b_m \rangle+ \langle b_l^\dagger b_m \rangle \langle b_l b_m^\dagger \rangle \end{equation} My question is when can I use this? In particular, I'm interested in finite temperature many-body perturbation theory and calculating the correlation functions from something like \begin{equation} Z[\overline{f},f] = \int \mathcal{D} (\overline{\phi},\phi) e^{-S_0 +S_{int}+\int_0^\beta d\tau \sum_l (\overline{f}_l b_l + b_l^\dagger f_l)}, \end{equation} where $S_0$ is the unperturbed part of the system, $S_{int}$ is the perturbation, and the final part of the exponential allows us to calculate the correlation functions via functional derivatives. Are there any circumstances under which I would need to calculate the four point correlation function itself? Or can I always use Wick's theorem?


title: Weak interaction and Neutrino

text: I have heard that Neutrino is the only particle affected by the Weak Interaction. How is a Neutrino affected by Weak interaction? I would like to ask the detail. Are there any relationships between WIMPs for the interaction?


title: How does a unique electron probability distribution correspond to one wavefunction?

text: I'm reading the Wikipedia article on DFT, and it says that there is a one-to- one correspondence between the ground state particle density $$n_0(\vec{r}) = N \int \text{d}^3 r_2 \int \text{d}^3 r_3 \cdots \int \text{d}^3 r_N \Psi_0^{*}(\vec{r}, \vec{r}_2, \ldots, \vec{r}_N) \Psi_0(\vec{r}, \vec{r}_2, \ldots, \vec{r}_N)$$ and the ground state wavefunction $\Psi_0(\vec{r}, \vec{r}_2, \ldots, \vec{r}_N)$ such that $\Psi_0$ is a functional of $n_0$: $$\Psi_0 = \Psi[n_0]$$ I don't understand how this is possible. It seems to me that you could have an infinite number of wavefunctions, with differing phases, as long as the squared amplitudes end up the same. Could someone explain to me why this isn't possible?


title: A modified version of the famous double-slit experiment

text: AFAIK, all the double slit experiments that were performed uses a light source (or electron source...) that emits photons at a _"perpendicular"_ angle as this image shows: (will call it experiment 1) ![experiment 1](http://i.stack.imgur.com/Z5Bb4.png) Now what would happen if the emitter was inclined (theoretically) as this diagram shows? In the sense of how it differs from the previous pattern obtained in experiment 1. ![experiment 2](http://i.stack.imgur.com/MTCqZ.png) Are there any experimental results that confirm this theoretical result?


title: Fundamental group of Calabi-Yau 3-fold in string theory

text: In string theory, we compactify a 10-dimensional space by a Calabi-Yau 3-fold to reduce the dimension to 4. To get a reasonable theory, a Calabi-Yau 3-fold should satisfy some properties. One is the Euler number must be $\pm6$ so that it is compatible with the generation of the elementary particles. I heard that the fundamental group of the Calabi-Yau 3-fold should not be trivial. What should it be?


title: What are the physics challenges of rain farming

text: Rain farming is an engineering concept, employing the technology of industrial cooling towers, to deliver vast amounts of water to the lower levels of the troposphere as warm moist air. This question is about the challenges to such an idea that might be presented by physics. The objective of rain farming is to deliver predictable rainfall to the region, for the benefit of agriculture and municipal hygiene. While the cloud physics are complex, there is a well documented history of tall cooling towers generating, usually unproductive, clouds. An additional objective of rain farming is to normalize regional hydrology, and to increase the stock of fresh water for the region by sourcing the water, for uplift to the troposphere, from briny sources such as oceans, non-potable underground resources, or flood plains. What constraints would be imposed on such an endeavour by the energetics of evaporation and condensation, atmospheric dynamics, etc.?


title: What is the transmittance function of a photo filter?

text: The effect of some photo filters can be summarized in transmission curves such as this one: ![http://micro.magnet.fsu.edu/primer/lightandcolor/colortemperatureintro.html](http://i.stack.imgur.com/xDw8d.jpg) I do not understand which physical value is modified by the value labeled as "Diffuse Density" in the picture above. It sometimes appear as a percentage, or a ratio included between 0 and 1. _Note: I originally posted a related question onphoto.stackexchange.com_


title: What is the theoretical upper limit on the rigidity of a material?

text: Take a perfectly rigid metal rod of length $\ell$ and some uniform linear density. Place one end at $(0,0)$ and the other at $(0, \ell)$. Over some reasonably short time interval $t$, perhaps on the order of a fraction of a second, displace the south end of the rod eastward from $(0,0)$ to $(1,0)$, while at the same time applying enough torque that the north end is displaced from $(0,\ell)$ to $(1, \ell)$. In practice it's very easy to do this so that the entire rod moves one unit eastward. But this is actually a classical view of the situation. To see this, make $\ell$ very long, say on the order of ten light-seconds, and large enough to be bigger than $t\cdot c$. Then at time $t$ the south end of the rod is at $(1,0)$ but the north end is still at $(0,\ell)$, because the north end can't have noticed yet that the south end has moved. But this has an implication for the material properties of the bar. I claimed in the first paragraph that it was perfectly rigid, but it now appears that it isn't as rigid as all that. Purely from speed-of-light considerations we can conclude that even a perfectly elastic bar must temporarily deform in the process of being translated from $x=0$ to $x=1$. It seems to me that if one assumed that the rod had length $\ell$ and uniform linear density $\rho$, then one could calculate the amount of force required to translate it from $x=0$ to $x=1$ by pushing on one end. Then supposing that the rest of the rod followed as quickly as speed-of-light propagation allows, one could calculate the stiffness of the rod, and this would be a theoretical upper bound on the maximum stiffness of any material whatever. But I don't have enough expertise or understanding of materials calculations to do actually perform this one. Also I suspect I must have left out something important, for the same reason. My questions are: >   1. Can this calculation be done, or is there some reason the whole idea is > unsound? >   2. If it does make sense, what upper limit on material stiffness does this > method produce? > I suppose that if it does work, the upper bound is vastly greater than the stiffness of any real material, but I don't mind that. (I found the question Extended Rigid Bodies in Special Relativity, which is clearly related to this, but doesn't get at what I want. My earlier question Behavior of shock waves at relativistic speeds started out as an attempt to ask this one, and somehow went in a completely different direction by the time I posted it.)


title: Solving Systems of Partial Trace Equations

text: Say I specify a quantum state - pure or mixed - by its partial traces on various subsystems. To what degree could one recover the original state, and what are the known methods for doing so? For example, consider the following partial traces on the state $\hat{\rho}$ of a system of two qubits, $a$ and $b$ $\rm{Tr}_a(\hat{\rho}) = \frac{\hat{\mathbb{1}}_b}{2}$ $\rm{Tr}_b(\hat{\rho}) = \frac{\hat{\mathbb{1}}_a}{2}$ One solution is the state $\frac{1}{\sqrt{2}}(|00\rangle + |11\rangle)$, but any state related to this one by unitaries on the individual qubits is also a solution. Do there exist general methods for either finding or characterizing the uniqueness of solutions to such a system?


title: Calculating angular velocity after collision

text: Suppose I have a disc which doesn't move, just rotate around the axis going through its centre of mass perpendicular to its surface. The disk has a stick perpendicular to its surface at the edge. I also have a block. The block hits the stick tangentially with certain velocity and then stops. How can I calculate the angular velocity of the disc after collision.


title: Are high redshift masses corrected for relativistic mass dilation? They would appear more massive right?

text: A distant quasar would be less massive in its frame of reference than our observations would suggest. Are such highly red-shifted objects corrected for relativistic mass dilation?


title: Can light be trapped theoretically?

text: My question is a little silly I know, but I'm curious to know if a particle of light can be theoretically trapped between two reflective screens. For example once the particle of light has left it's source, and is reflected against a mirror (assuming there is no loss of energy) and you immediately replace the source of light with another mirror. Would the light beam bounce in between the sheets endlessly?


title: How does one calculate the force applied on an object by a magnetic field?

text: I've tried very hard to find an answer to this question, and every path leads me to an abstract discussion of fundamental forces. Therefore, I will propose two very specific scenarios and see if they yield the result that I am looking for. # Scenario One Let's say that I have a vertical tube exactly 1 inch in diameter that is completely incapable of holding an electromagnetic charge and has a frictionless surface. Resting inside this tube is a steel ball also exactly 1 inch in diameter. If a cylindrical magnet, also exactly 1 inch in diameter is slowly lowered into the tube, how does one determine the exact point at which the force being applied to the steel ball by the magnet will cause the ball to overcome gravity and rise toward the magnet? Is there even any way to determine this? What further information would I need? # Scenario Two I have the same tube from above with the cylindrical magnet resting on the bottom of the tube, north pole facing upward. Suspended by a weightless string in the tube is an identical magnet, north pole facing downward. If the bottom magnet is slowly raised, how does one calculate the exact point at which the suspended magnet will begin to move upward? Is this calculation possible? What further information would I need for this calculation? # Extra Question How are weight capacities on magnets calculated? I.e. if a whitepaper says that a magnet is capable of lifting 25 pounds, how is the correct size magnet calculated?


title: Would a matter-antimatter explosion cause fallout?

text: I know matter and antimatter annihilation release a lot of gamma rays which are considered ionizing radiation if I am not mistaken. But what if the explosion happened on the surface of the earth, would the the material taken into the fireball cause fallout afterwards ? My question in another form, what causes fallout ? and is ionizing radiation capable of radiating materials for a long time ?


title: Is thermodynamic free energy and potential energy the same thing?

text: The equation for free energy $F$ and potential energy $E_{pot}$ are: $$ F=U-TS \\\ E_{pot} = E_{tot} -E_{kin} $$ But the temperature $T$ is proportional to the average kinetic energy of a system. So the equations are really similar in case the entropy is constant. In addition the potential energy is sometimes defined as _capacity of a system to do work_. On the other side free energy is defined as _the amount of work that a thermodynamic system can perform_. Isn't that the same thing?


title: Peaks and dips position in the Planck CMB

text: I am currently looking for the values of the firsts peaks and dips (at least the first 3) in terms of l and angular size for the CMB angular power spectrum measured from Planck. But the only thing I can find is the plot, not the values. What are these values ?


title: Conserved current for a constant translation of a free massless scalar field

text: In Zinn-Justin's _Quantum Field Theory and Critical Phenomena_ they start with an action for a free massless scalar field: $$S(\varphi) = \frac{1}{2}\int d^{2}x\left[\partial_{\mu}\varphi(x)\right]^{2}$$ And say that the action is invariant under constant translations of the field $$\varphi(x) = \varphi^{\prime}(x) + \theta$$ This is all fine, the problem I'm having is with the next part. They say a conserved current $J^{V}_{\mu}$ corresponds to this symmetry: $$J^{V}_{\mu} = \partial_{\mu}\varphi(x)$$ I just can't see how to get that out with $\theta$ being a constant - unless I'm missing something in the notation.


title: Increase path length for a photon with two mirrors vs optical fiber

text: Is it possible to build two mirrors in vacuum such that a light pulse (single photon) gets refleted many times in order to increase the distance it travels? For example like that? ![enter image description here](http://i.stack.imgur.com/Tm14U.png) Say I want the photon to travel 1 km. The two mirrors have a length of 1m and a distance of 10 cm. Hence I need the pulse to bounce back and forth approximately 10,000 times. Is it possible to build something like that? Another question is: What happens to the pulse? Will it disperse? How will the incident pulse differ from the outgoing pulse? And at least: Is there any advantage over an optical fiber with a length of 1 km? Or is it even worse (concerning losses and dispersion)?


title: Does the the quantum field theoretic process of particle–antiparticle annihilation break the axioms of Special Relativity?

text: ![Feynman diagram](http://i.stack.imgur.com/I6pKL.png) $\textbf{Note that this diagram hasn't anything to do with the question directly.}$ After a particle and its antiparticle annihilate, their energy is converted into a force carrier particle, such as a **gluon** , **$W$ or $Z$ force carrier particle** or a **photon**. When I contemplate this phenomenon, it seems curious to me that a system can have a mass, and then accelerate to the speed of of a massless particle. This curiosity stems from the special relativistic notion that it takes an infinite amount of energy to make any particle with mass accelerate to the speed of light. The equation that I am speaking of is graphed here, and demonstrates the well recognized $\textbf {Energy-Mass against Velocity}$ asymptote of Special Relativity: ![enter image description here](http://i.stack.imgur.com/jfaRu.jpg) Besides the fact that this process adheres to the many other conservation laws which we have discovered, is there a reasonable explanation for this phenomenon? Is it just like putting a value into a function, and the universe knows to produce a photon (or another moderator)? Or, does something else take place, where the particles are exposed to energy in its purest form, so they accelerate tremendously together until they separate to form another particle pair?


title: Fermion Self-Interaction

text: I'm trying to think of a theory with a Fermion self-interaction, similar to the $\phi^4$ theory. The first difficulty is of course that such a theory would have a non- renormalizable mass dimension: for example, $[(\bar{\psi}\psi)^2]=mass^6$, so to get to $mass^4$ you need a coupling constant with $[g]=mass^{-2}$ which is not-renormalizable using the formula $$SDoD=4-\sum_{f}E_{f}\left(1+s_{f}\right)-\sum_{i}N_{i}\left[g_{i}\right] $$ where $E_f$ is the number of external legs of field $f$, $s_f=0$ for Bosons and $\frac{1}{2}$ for Fermions, $N_i$ is the number of vertices of operator of type $i$ and $g_i$ is the coupling constant of operator of type $i$. Then I started wondering, why does the Fermion field _need_ to have a mass dimension of $\frac{3}{2}$? Because of the term $\bar{\psi}\partial\psi$. So what I'm wondering about is: is there any way to fix this and get a renormalizable theory? Is it worthwhile to study Fermion self-interactions even for non-renormalizable theories? (non-renormalizable theories could still be predictive, as effective theories?)


title: Regarding real field Klein Gordon Equations

text: Here are 2 doubts:   1. If we change the sign of the mass term in the free massive KG Lagrangian to get: $L = \frac{1}{2}\partial^\mu\phi\partial_\mu\phi + \frac{1}{2}m^2\phi^2$, What would be the $physical$ implications of this change? (aside from on shell condition not being satisfied)?   2. Let $\phi^{(1)}$ and $\phi^{(2)}$ be 2 real scalar fields with the Lagrangian: $L = \frac{1}{2}\sum_{i}\partial^\mu\phi^{(i)}\partial_\mu\phi^{(i)} - \frac{1}{2}m^2 \sum_{i,j,k} \phi^{(i)} M_{ij}M_{jk} \phi^{(k)}$ . where $M_{11} = \lambda$, $M_{12} = 1$, $M_{21} = 1$, $M_{22} = 0$ AND $\lambda >> 1$. What is the mass ratio of the 2 particles in the theory? EDIT: As for the 2nd question, I found that $L$ should simplify to $\frac{1}{2}\sum_{i}\partial^\mu\phi^{(i)}\partial_\mu\phi^{(i)} - \frac{1}{2} m^2(\lambda \phi^{(1)} + \phi^{(2)})^2$. I tried continuing from the viewpoint that we can consider the superposition $\phi^{(3)} = \lambda \phi^{(1)} + \phi^{(2)}$ and try to eliminate cross-terms, thereby getting a simple sum of non-interacting Lagrangians, but that didn't work. (Now I'm out of my depths) Thanks in advance


title: How Fundamental is Spin-Orbit Coupling to Topological Insulators?

text: I'm well aware this is a very active area of research so the best answer one can give to this question may be incomplete. Topological states in condensed matter are well-known, even if not always recognized as such. The most famous example is likely the quantum Hall effect. In this case, time-reversal symmetry is broken by an external $\vec{B}$ field. In the past decade, it was realized that spin-orbit coupling can be used to break time-reversal symmetry as well. This leads to topologically preserved states in so-called topological insulators. However, I've overheard that some condensed matter theorists believe spin- orbit coupling may not be necessary for breaking time-reversal symmetry in topological insulators. Apparently, there are some other mechanisms proposed in which this breaking is not (or at least not primarily) due to spin-orbit coupling. I've heard from a fairly well-respected condensed matter physicist that he believed spin-orbit coupling was important in all realistic topological insulators, but probably not essential for the theory. Being a relative novice in the area, I don't know of any other mechanism by which time-reversal symmetry could be broken. Besides spin-orbit coupling effects, is there any other way that topologically protected states could exist with 0 $\vec{B}$ field? If so, how realistic are these? If not, what is meant when people claim spin-orbit coupling is not fundamental to topological insulators, and what would be a more fundamental way to look at it? Any references are certainly appreciated.


title: Modeling stochastic process with frequency-dependent power spectrum

text: I'm trying to model of Johnson-Nyquist noise propagation in a nonlinear circuit. An ideal (linear) resistor can be modeled very nicely by the Fokker- Planck equation (equivalently, the drift-diffusion equation), where charge $V/R$ flows through the resistor _on average_ , but there's also random flow of charge either way across the resistor characterized by "diffusion coefficient" $k_BT/R$. I get a nice differential equation describing how charge probabilistically flows through my circuit. Everything is good. Then stage 2 is to have a resistor with frequency-dependent resistance (like all resistors in the real world). Here I get stuck... A time-domain-based analytical solution seems impossible because---with frequency-dependent resistance---it would seem that charge transport across the resistor right now depends on the entire history of charge transport in the past. A time-domain numerical (monte carlo) solution seems impossible because ~~the relevant frequencies vary over many orders of magnitude~~ I don't know how to construct a time-domain stochastic model with a predetermined power spectrum. Any kind of frequency-domain solution seems impossible because other parts of the circuit are extremely nonlinear and therefore mix different frequencies together. Any advice? Am I missing some trick?


title: What is meant by potential energy for a particle in a field?

text: Potential energy is usually defined using a field and a particle that experiences the field force, as the work down in moving a unit particle from infinity to a position in that field. But some physics text books describe the particle placed there as possessing potential energy, others that the potential energy is "stored" in the field itself, which appear to conflict with one another. So what is the modern meaning of potential energy for a particle in a field?


title: Mass corrections to fermions proportional to the mass?

text: In this post regarding quantum corrections to a massless fermion field, the answerer stated that quantum corrections to the mass will always be proportional to the mass (at least in QED). This point leads to an important claim that a massless fermion remains massless under quantum corrections in the Standard Model. I can show that this is true to 1 loop order in QED, but is this always the case? If so then why?


title: What's the simplest method for accurately measuring the speed of light?

text: With common household objects in hand, and, say a budget of $20, what's the simplest experiment for accurately measuring the speed of light?


title: How to determine divergence of a LED source from a single biconvex lens

text: I'm trying to determine the divergence angle of light from a single lens that is completely illuminated by a high power LED. Most optics textbooks only deal with imaging optics and I'm having a hard time wrapping my head around what happens when I don't care about the image on the focal plane - this is for a non-imaging application (I'm just projecting the light), and I just want to be able to determine the spot size at any distance. The LED will not necessarily be at the focal point of the optic (though it will be along its axis). Furthermore, the LED isn't a point-source, as its size is approximately $\frac{1}{10}^{th}$ of the focal length of the lens.


title: Disproof of Bell’s Theorem

text: The half-page arxiv doc by Joy Christian of Oxford Uni, UK has the Title and Abstract: **Disproof of Bell’s Theorem** > We illustrate an explicit counterexample to Bell’s theorem by constructing a > pair of dichotomic variables that exactly reproduce the EPR-Bohm > correlations in a manifestly local-realistic manner. As The Bell's theorem is often cited it seems important to clarify this short doc.   Is the doc OK? This recent paper Comments on "Disproof of Bell's theorem" by Florin Moldoveanu (July 06,2011)   can shed light in the interpretation and importance of the work of Joy and may be the case that the answers could benefit from it. One conclusion that seems relevant, imo, is that Geometric Algebra is very promising to model the physical world. LM post @TRF pointed to a pre-print of Gill ( _Simple refutation of Joy Christian's simple refutation of Bell's simple theorem_ )   and the Joy answer ( _Refutation of Richard Gill's Argument Against my Disproof of Bell's Theorem_ )   (thanks LM)


title: if one electric heater makes the room 25 degrees, will two make 50?

text: The question in the title is of course childish and the answer is NO...   But then how much if not 50 degree ? lets give some more data:   we have a electric heater with **NO temperature regulation** and only one stage On or Off.   It is used to heat a small room and if the outside temperature is 0, the room temperature stays around 25. How will the room temperature change if we add another electric heater of the same type ?


title: Auto-ionization of water

text: > $HCl(g) + H_2O(l) -> H_3O^{+}(aq) + OH^{-}(aq)$ Assuming $HCl$ completely ionizes in water, if we prepare a $HCl$ solution, will the $[HCl]=[H_3O]$ ($HCl$ concentration = $H_3O$ concentration)? That is, the $HCl$ before the reaction, and the $H_3O$ after reaction? If so, then if we observe the following equilibrium reaction for the auto- ionization of water: > $2H_2O \rightleftharpoons H_3O^{+} + OH^{-}$ it can be concluded that $[H^{+}][OH^{-}]$ is a constant value, that is, the product of their equilibrium concentrations is a constant value. However, if the $H_3O$ concentration increases by the ionization of $HCl$, wouldn't the reaction shift to the left to counteract the change (by Le Chatelier's principle)? Consequently, this would decrease the concentration of $H_3O$ ions in solution, and contradict the equality of $HCl$ concentration = $H_3O$ concentration, so was the original proposition incorrect?


title: What is the difference between an electric and a magnetic field?

text: This question is a consequence of another question of mine which is about spin. Here is my spin question. What is the difference between these two fields? How do they occur? Am I right if I say that a magnetic field is about photons (because they occur between N and S poles of a magnet) and an electric field is about electrons? How they are related? Finally; when and why do we use the word "electromagnetism"?


title: Electric field intensity at the center of a small cavity in a large block of polarized dielectric

text: I'm having trouble with an exercise in my textbook. The question is: Determine the electric field intensity at the center of a small spherical cavity cut out of a large block of dielectric in which a polarization P exists. Since there are no free charges, the electric flux density D must be equal across the interface between the cavity and the dielectric. Additionally, the polarization vanishes inside the cavity. I also assume that the electric field intensity vanishes inside the dielectric body (is this valid?). Applying these equations we get: $$P = \epsilon E \implies E = \frac P \epsilon$$ (at the boundary) I assume the electric field intensity is unchanging through distance inside the cavity (inside the cavity there are no free charges and the polarization vector is zero and the flux across ANY closed surface must be zero $\implies$ E is constant). However, according to the textbook, this isn't the correct answer. Where did I go wrong? EDIT: I've seen another question (Electric Field from Dielectric Shell) which implies that the solution is that $E = 0$ at the center (since $P$ vanishes). The logic makes sense, but this answer is still different from the textbook.


title: Is Galilei's reasoning on free fall valid?

text: Galileo Galilei discovered by experiments that all bodies tend to fall with the same rapidity (I use it in an intuitive sense, you can replace it by 'acceleration' used in today's physics language), independently of their weight. He also provided the following rationale, based on the proof by contradiction (I do not have the original wording at hand, but I believe I can paraphrase the idea, which I am interested in.) Let's imagine one heavy body and one light body. Suppose that heavy bodies fall faster than light bodies, as almost everybody believes. Connect the two bodies so that they form another body. This resulting body is heavier than the original heavy body, so according to the assumption it should fall faster. But on the other hand, the original heavy body is being inhibited in its fall by the connected light body, since this "wants" to fall with less rapidity. Due to this inhibition, the heavy body part should fall with less rapidity than it falls normally alone. So we arrive at contradiction, and the only way to resolve it is to reject the assumption. Instead, all bodies fall with the same rapidity (acceleration). What do you think of this argument - is it valid, or no? For what reasons? It is very compelling at first, but on the other hand, should not the law of free fall be experimental law, rather than logical necessity? If so, where is the problem with the reasoning?


title: Mech stability through gyroscope

text: I recently read up about gyroscopes, angular momentum and mechs (the big Cockpit controlled robots) and was wondering if it would be possible to get a stable walking mech (only as example, not meant to be a serious attempt http://autopixx.de/bilder/lM23mgzq/battle-mech.jpg) by adding a big enough gyroscope which stabilizes the mech as it lifts a leg in the walk cycle. The mech is about 4 meters high, but I can't speculate on how much it probably would weight. Just a question out of curiosity. Thanks in advance :)


title: How does $SU(2)$ group enters quantum mechanics?

text: What is the reason that $SU(2)$ group enters quantum mechanics in the context of rotation but not $SO(3)$? What really rotates and which space it rotates? It cannot be the physical electron that rotating in real space. I think it is the state vector that rotates in spin space. Am I right? Is it the property of the "weird" spin space (Hilbert space) that rotation by $4\pi$, brings it back to where it started?


title: How to apply conservation of angular momentum with a shock?

text: I got this tricky question, need help. A uniform rod of mass $M$ and length $L$ is attached to an axis at its top, a bullet with mass $m$ traveling at speed $U$ (horizontal) hits the rod at $2L/3$ from the axis and then LEAVES the rod at speed $U/2$. I need to find the angular speed of the rod right after collision assuming collision time is very short. If the bullet stayed in I would just do conservation of angular momentum, find the new moment of inertia with the bullet using Steiner's law and solve for $\Omega$. But what do I do when the bullet leaves the rod?


title: Current, Current density

text: edit: Hi I'm trying to find the magnetic field generated by a time dependent oscillating current in the quasistatic case ($|z|,r <<c\omega$) where r is the perpendicular distance from the z-axis. The current is flowing through a long, thing wire that is laying on the z-axis. If $z=0$, we can write the current as $$ I(t)=I_0\sin \omega t $$ and now I am trying to find the magnetic field $B(r,t)$ at $z=0$ with this current. How can we calculate $\vec B$? Possibly, can we calculate $\vec J$ to calculate $\vec A$ to calculate the magnetic field? THanks a lot for your help on the last problem too.


title: Rope on an inclined plane problem

text: ![enter image description here](http://i.stack.imgur.com/DkxmN.jpg) My book says the answer is (a)zero but i don't understand how it came zero. What will the acceleration if horizontal level of the two ends of the rope are different?


title: What is the frictional force of this car?

text: The breaking acceleration for a car is $-12.2\frac{m}{s^2}$. The car has a mass of 925kg. I want to find the frictional force. Here is what I have so far: $$\Sigma\vec{F} = m\vec{a}$$ $$\Sigma\vec{F} = (925kg)(-12.2m/s^2)$$ $$\Sigma\vec{F} = -11285N$$ So now I have the net force of the car. Would this be the same as the frictional force (any action has an equal and opposite reaction)? Or do I need to use the Friction Law formula? ${}{}$ Thanks!


title: Instantons in Witten's supersymmetry and Morse theory

text: I'm reading Witten's paper on supersymmetry and Morse theory and am confused about the details of the instanton calculation which he uses to define a Morse complex (beginning at page 11 of the pdf) . Witten writes down the relevant supersymmetric Lagrangian and then states > Instanton solutions or tunneling paths in this theory would be extrema of > this Lagrangian, written with a Euclidean metric and with the fermions > discarded. I get the part about the Euclidean metric, but why discard fermions? After finding the instanton solutions (which are just paths of steepest descent with regard to the Morse function. We are interested in paths connecting critical points.), he then proceeds to state that for quantum fluctuations around the classical solution, > Nonzero eigenvalues cancel between bosons and fermions, due to > supersymmetry. We are left with the zero eigenvalues of the fermions. For a > trajectory running from A to B, the index of the Dirac operator equals the > Morse index of A minus the Morse index of B. How does this cancellation between fermions and bosons work? What about the zero eigenvalues of the bosons? The Dirac operator here is just exterior derivation (or more precisely, the perturbed version thereof), correct? But what is meant by "the Dirac operator for a trajectory from A to B"? A large part of the problem is that nearly all references I could find connecting Instantons and supersymmetry deal with Yang-Mills theory, or extended supersymmetry in far more complicated settings (My background in QFT is rather limited). Even then, I have never found anything which suggests "discarding the fermions" in the instanton calculation. I mostly understand the rest of the paper, but this part leaves me mystified.


title: Can friction be brought in to zero value by means of laboratory experiments?

text: By newton's first law, an object will continue to be in the state of motion of rest until and unless an external force is acting on it. To prove or verify it experimentally today, is there any way out to diminish the problem of friction?


title: How does acceleration affect the orbit?

text: I want to know what roughly happens if we hit an orbiting body a little. And how is it possible that most of the objects have seemingly circular orbits around. How do they stabilize? I ask because it seems to be that celestial objects must recede from their orbits. The orbit is simply a distance r from the massive object and it depends on the speed v of the orbiting body. We know that $F=ma=mv^2/r$ and gravitational force is $mMG/r^2$ so that speed of orbiting must be $v_o^2=MG/r_o$ or $v_o=\sqrt{MG/r_o}$, which implies kinetic energy of $E_{kin} = mv^2/2 = mMG/2r$. Also, the binding energy is $U = mMG/r_o = E_{kin}/2$. That is fine. It says that the kinetic energy of our orbital motion is a half of binding energy. That is, our speed is lower than the escape velocity, which from the $mv_e^2/2 = mMG/r$ is $v_e = \sqrt{2MG/r_o} = \sqrt 2 v_o$, twice the orbiting speed. I guess that we will fall onto the Earth surface if move lower than $v_o = \sqrt{MG/r_o}$ and escape the Earth into infinite space if move $\sqrt 2$ faster than that. My question is basically, **what happens if we move at the speed between orbiting speed and escape velocity, what would be the orbit**? Furthermore, I wonder how do objects stabilize at their natural orbits. Suppose we have accelerated our naturally orbiting object at speed $v_o$ k times. This means that new speed is $v_2 = k v_o$ and radius must be reduced since orbiting speed and radius are inversely proportional, $r_2 = MG/v_2^2 = r_o/k^2$. However, obviously, accelerated body will not tend towards the attractor, it will tend to escape it. That is why I wonder. Satellite, accelerated from its $v_o$ to $v_2$ at height $r_o$ will raise up but loose as getting higher. As computed above, $U_{binding} = 2 E_{kinetic}$ so $E_{total} = {3 \over 2} mv_o^2$ and (here, $v_{on}$ stands for natural speed at orbit n) we have $$E_{total2} = {3\over 2} mv_{o2}^2 = E_{total1} = (U = mv_o^2) + E_{KinAcelerated} =\\\ mv_o^2 + {m\over 2} v_2^2 = mv_o^2 + {m \over 2} k^2 v_o^2 = mv_o^2(1+{k^2 \over 2})$$ whence speed at the raised orbit will be $v_{o2} == v_o \sqrt{{2 \over 3} (1+k^2/2)} = v_o \sqrt{(k^2 +2)/3}$ That is, if we accelerate the satellite 1 times, $v_0 \rightarrow v_0 $ then the orbit will not change, $v_{o2} = v_o \sqrt{(1 +2)/3} = v_o$. Voila! Increasing the speed $\sqrt{2}$ times, up to $v_e$, will make $v_{o2} = v_0 \sqrt{4/3}$, which corresponds to the orbit $r_2 = 3 MG / (4 v_o)$. It did not gone into infinity. It was reduced by 4/3 with respect to $r_0$. We declined instead of raising the orbit and speed has accelerated! Again, the theory says that satellite must reduce the height when it is accelerated. But, I do not believe that this is what happens in reality. In reality in should go higher and even higher because despite of loss of the speed, attraction force is also reduced at higher heights. On the other hand, if most of the observed satellites orbit at their natural orbits it means that the formula is right and accelerated satellite will go down rather than raise higher. I assume that it is not important for the orbiting and escaping speeds if our speed is horizontal or vertical. I guess that absolute value is only important and do not know how to make the computations otherwise.


title: Verifying Newton's 2nd Law

text: **Background/Experiment setup** : In one of my classes a mass (0.04 kg) was hung from a pulley and attached to a much heavier mass (around 0.4 kg) that rest on an airtrack. A thread connected the hanging mass with the one on the track. The hanging mass would be dropped and the acceleration of the mass on the airtrack would be calculated. For 5 trials the mass was increased .01 kg and the acceleration calculated. Then I divided gravity by each calculated acceleration so I ended up with 6 new numbers. Then for each mass I divided $m_1$ ($0.4$ kg) by each hanging mass. Then I plotted $\frac{m_1}{m_2}$ as the $x$ axis of a graph and $\frac ga$ as the $y$ axis. Then I found a line of best fit: $y = 1.0004x + 0.99 $ $r^2 = 0.9994. $ So it produces a very straight line, the data line up very well and produce a great line of best fit. **Question** : How does this verify Newton's 2nd law? I understand that it is $F=ma$, and the graph does show that as mass increases so does acceleration proportionally. IS all that is needed to verify it? I'm having a hard time in my lab write up actually explaining how this set up verifies Newton's 2nd law. Thanks for reading!


title: Magnitude of magnetic field at the center of circular wire

text: I'm preparing for an exam by solving the sample questions , here is the one I'm having difficulty with : Following is the given circuit.   Which contains two resistance $R_1$ and $R_2$ in form of circle of radius $r = 1 m$ with a battery having e.m.f. $V = 10π $ volt.   Upper resistance is having resistivity $p_1 = 4$ & lower resistance having resistivity $p_2 = 2$.   Angle between two points $A$ and $B$ is $60°$. (wires have same cross section $A_1 = A_2 = 2 cm^2$) Diagram : ![Circuit](http://i.stack.imgur.com/5MiAz.png) Here is the question based on this passage that I'm having difficulty with :   Find magnitude of magnetic field at center. (A) 0T (B) 1T (C) 2T (D) πT The problem is I don't know any formula to calculate the magnetic field at a point nor am I expected to know (probably , at least it is not in syllabus) .   Also what is 'T' ? Is it unit of magnetic field?   The answer is 0 so I guess I should be able to reason why it should be 0 but I don't know how. Please help.


title: The temperature dependence of the electron-hole (or particle-hole) continuum

text: In the theory of the electron gas, the particle-hole pairs are possible elementary excitations. I would like to know how the temperature T affects the particle-hole continuum which defines the domain where the imaginary part of the density-density response function (or Lindhard function) is different from zero. I need to understand it in order to calculate the Lindhard function at finite temperature.


title: Is radioactive decay spontaneous or random?

text: When the count rate of a radioactive isotope is measured, the readings fluctuate. Do the fluctuations demonstrate the random nature of the decay or the spontaneous nature? (This question was asked in an exam. I don't get what the difference is between random and spontaneous here.)


title: Eigenstates of half Harmonic Oscillator

text: This might be a stupid question so pardon me! If I am looking for energy eigenstates to the 1D quantum problem such that there is an infinite barrier at $x<0$ and for $x>0$ the potential is $kx^2$, what would it be like? Please prove that your set of eigenstates forms a complete set.


title: Noether current for a local gauge transformation for the Klein-Gordon Lagrangian

text: The Noether current corresponding to the transformation $\phi \to e^{i\alpha} \phi$ for the Klein-Gordon Lagrangian density $$\mathcal{L}~=~|\partial_{\mu}\phi|^2 -m^2 |\phi|^2$$ by finding $\delta S$, and setting it to zero. The general formula for a global transformation is $$j^{\mu}=\frac{\partial \mathcal{L}}{\partial(\partial_{\mu} \phi)}\Delta \phi-\mathcal{J}^{\mu},$$ where $ \partial_{\mu} \mathcal{J}^{\mu}$ is the change in the Lagrangian density due to the transformation. (See Peskin section 2.2). **How do I find the Noether's current corresponding to a local transformation $\phi \to e^{i\alpha(x)}\phi$?**


title: Magnetic Susceptibility at Arbitrary Temperature

text: I'm currently working on an assignment where the questions is: > Consider a gas of N noninteracting electrons in a uniform magnetic field > **B** = B$\hat{z}$ in a macroscopic system. Assume that the field affects > only spin of the electron and not its orbital motion, and denote the Bohr > magneton by $\mu_{B}$. Show that the magnetic susceptibility $\chi$ at > arbitrary temperature is given by $$\chi = 2\mu_{B}^2 \int d\epsilon > \frac{dD(\epsilon)}{d\epsilon}f(\epsilon)$$ where $D(\epsilon)$ is the > density of orbital states and $f(\epsilon)$ is the Fermi-Direc distribution > function. Express $\chi$ in terms of $N$ and fundamental temperature $\tau$ > in the high-temperature limit, assuming that the lowest single-particle > energy is zero and $D(0) = 0$. Now, in a previous assignment, we already had the question: > Consider a gas of N noninteracting electrons in a uniform magnetic field > **B** = B$\hat{z}$ in a macroscopic system. Assume that the field affects > only spin of the electron and not its orbital motion. Find the spin magnetic > susceptibility, $$\chi = \frac{dM}{dB}$$ where $M$ is the total > magnetization of the system. Denote the magnitude of the magnetic moment > associated with the electron spin and charge as $\mu_B$ (Bohr magneton). And the solution to that question was > $\chi = 2D(\epsilon_F)\mu_B^2$ Clearly quite similar to the desired function for the newer question. Now, I've been looking at this question for two days. I've gone over the solution to the old question numerous times, but I don't know where to start for the second question - I don't know at what point the arbitrary temperature consideration becomes relevant and how it changes the analysis. I am by no means looking for a solution to the question, but I am hoping that somebody could give me a pointer or two to get me going in the right direction. At first glance, I suspect the solution has something to do with the equations $$ N = \int_0^\infty d\epsilon D(\epsilon)f(\epsilon)$$ (as opposed to the equation $$ N = \int_0^{\epsilon_f} d\epsilon D(\epsilon)$$ which was used in the previous assignment), and $$ M = \mu_BN$$ But I don't know how to get from these to the final result. Magnetic susceptibility is a concept that both my class's notes and my textbook don't go into much detail on. Like I said, any hints or pointers would be appreciated.


title: Loss of Power at high frequencies

text: One of my work colleagues told me that a cable he is sending a signal through is losing power at high frequencies. So he recommends the signal should be amplified before being sent. The explanation was given for the power loss is that higher frequency signals are lossier. As a newcomer to signal processing, I'd like to understand more about why or how this effect occurs. How is the frequency of the signal causing the power loss to occur? From Wikipedia's article on Coaxial cable I found this, which seems promising: > If an ordinary wire is used to carry high frequency currents, the wire acts > as an antenna, and the high frequency currents radiate off the wire as radio > waves, causing power losses. Is understanding how antennae work key to understanding why the high frequency results in power loss?


title: Does gravity require strings?

text: OK, before I ask my question, let me frame it with a few (uncontroversial?) statements:   1. The low-field-limit plane-wave solution to Einstein's equations is helicity-2.   2. In the early days of string theory, when people were factorizing the Veneziano amplitude and its generalizations in order to find a spectrum of physical states, they could never get rid of the massless spin-2 state (i.e., closed loop) that kept cropping up, although they tried mightily to do so.   3. In string theory, one can only formulate a physical theory of interacting massless spin-2 states (i.e., closed loops) if the gauge symmetry of those states can be deformed into the diffeomorphism symmetry of GR.    4. Conformal invariance of the 2-D worldsheet action of a closed loop in string theory leads via state-operator correspondence to gravitational-wave solutions of GR. Points (3) and (4) were shamelessly lifted from Lubos Motl's blog. All this goes to say is that string theory "postdicts" gravity, as Ed Witten put it in a talk I saw back in the 90s. In other words, it seems that as soon as one starts writing down a quantum theory of interacting strings, gravity drops out of it like magic and there's nothing you can do about it. This brings me to my question. If a closed loop is intrinsically spin-2 and by its very nature must satisfy the symmetry requirements of general relativity, why can't one work backwards from these seemingly stringent symmetry requirements and demonstrate that a massless spin-2 point particle can NEVER satisfy them? Is it possible that only an extended object can serve as the carrier of a force which manifests itself as the curvature of spacetime? I would think that such a demonstration would show that not only does string theory postdict gravity, but gravity requires strings.


title: Question about Matrix Integral

text: I am stuck with the technique details of KKN's paper . How to get formula (2.11) $$Z= \int \frac{d \phi }{ \rm{ Vol(G)}} \frac{1}{\rm{Det}(\rm{ad}(\phi)+\epsilon)} $$ from (2.10)? $$Z=\int \frac{d \phi dX_a d\psi_a}{ \rm{ Vol(G)}} \exp {\kappa_1 \rm{Tr}( i\phi[X_1,X_2] -\frac{1}{2}\epsilon^2(X_1^2+X_2^2)+\psi_1\psi_2) } $$ I have no idea about this calculation and I even do not know the meaning of $\rm{Det}(\rm{ad}(\phi)+\epsilon)$. Any advices are welcoming.


title: Why does a thermometer in wind not show a lower temperature than one shielded from it?

text: I'm a little familiar with the physics and thermodynamics of the wind chill effect, but this question seems to come up from time to time: Why, given two temperature sensors or thermometers in the same environment, do both report the same temperature if one is exposed to wind when the other is shielded from it? People often ask because the temperature reported by, for example, their vehicle, doesn't seem to change as they drive at different speeds, etc. (Other than of course the changes from one actual temperature to another as they change geography.) My understanding is that inert devices aren't endothermic like we are, so the effects of wind chill don't apply. Can you explain this in layman's terms?


title: What is quantum discord?

text: What is quantum discord? I stumbled upon this term on Quantum Computing: The power of discord, but have never heard of it before. Can you give a bit more mathematical explanation of the term here?


title: What theories allow for "Tractor Beams"?

text: NASA recently announced $100k grant to study the possibility of using laser tractor beams. What tractor beam concepts have been published but not been dis-proven?


title: Neon lamp: minimal breakdown voltage

text: I am looking at this formula from wiki for breakdown voltage in gas discharge lamps, ~~and I see its linear by length (d)~~ (oops, I see it's divided by $ln$) $$\frac{Bpd}{\mathrm{ln}\ Apd-\mathrm{ln}\ (1+\frac{1}{\gamma_\text{se}})}$$ Does that mean that I can make tiny neon lamp with just 0.1 to 0.05mm spacing between electrodes and it will glow as low as 5V for example? Are there any gases which have lower minimal breakdown voltage than neon? I guess using niddles instead of parallel plates would also lower breakdown voltage, as field is stronger?


title: Two-level system with spontaneous emission

text: I have been stuck with this elementary problem of two-level system including spontaneous decay. After solving by standard procedure the following pair of equations, when I plot the expressions of the ground state population $P_0 (=|C_0(t)|^2)$, excited state population $P_1 (=|C_1(t)|^2)$ and sum of the two, $P_0+P_1$ vs time $t$, I find that $P_0+P_1$ is not $1$, as if there is a loss of the system as a whole. It has got to be wrong. I tried to solve the following coupled differential equations incorporating the spontaneous decay width, for a single two-level atom subjected to optical field: $ \dot{C_0(t)} = -\frac{i}{2} \Delta C_0(t) + \frac{i}{2}(\Omega-i\Gamma_{sp})C_1(t) \\\ \dot{C_1(t)} = \frac{i}{2} (\Delta+i\Gamma_{sp}) C_1(t) + \frac{i}{2}(\Omega^*)C_0(t) $ Is it correct to incorporate spontaneous emission in the language of probability amplitude of a single atom like this?


title: What is / are the primary criticism(s) against Einstein-Cartan-Evans field theory?

text: What is / are the primary criticism(s) against Einstein-Cartan-Evans (ECE) field theory? On Wikipedia the references provided were: arXiv:physics/0607186, MR2372785 (2008j:83049b), MR2218579 (2007a:83003), MR1832162 (2002d:78002). I'm searching for an undergraduate-level* explanation of the criticisms / details. (The first link is openly accessible [i.e. arXiv], but the others are from MathSciNet; (as I don't have access to MathSciNet, I'm wondering if someone with access may provide their take on the contents.)) (*To be specific, I have background in the following courses: Physics: First- year Mechanics, Basic E&M (but not electrodynamics); Math: Calculus I, Linear Algebra, some Differential Equations.)


title: Why don't experimental physics groups have statisticians in it?

text: Perhaps someone can clear up a bit a cognitive dissonance I am experiencing. Pollsters are under constant scrutiny of statisticians for even the most mundane of survey topics. With so much riding on the results of fundamental physics experiments, why don't we _need_ statisticians to do the data analysis for us (or at least be looking over our shoulders)?


title: How far can you get (in quantum mechanics) with just commutation relations?

text: Clearly it is possible to derive a set of commutation relations from some Hamiltonian, and certainly they give useful and interesting invariants when investigating the behavior of quantum systems. But my question is can we go backwards? In other words, can one set up a collection of (anti)commutation relations which determine a particular C* algebra, all of whose representations are solutions to some Schrodinger equation? And if not, how much extra data do we need to reconstruct the exact Schrodinger equation?