Why quantum mechanics?

Imagine you're teaching a first course on quantum mechanics in which your students are well-versed in classical mechanics, but have never seen any quantum before. How would you motivate the subject and convince your students that in fact classical mechanics cannot explain the real world and that quantum mechanics, given your knowledge of classical mechanics, is the most obvious alternative to try? If you sit down and think about it, the idea that the state of a system, instead of being specified by the finitely many particles' position and momentum, is now described by an element of some abstract (rigged) Hilbert space and that the observables correspond to self-adjoint operators on the space of states is not at all obvious. Why should this be the case, or at least, why might we expect this to be the case? Then there is the issue of measurement which is even more difficult to motivate. In the usual formulation of quantum mechanics, we _assume_ that, given a state $|\psi \rangle$ and an observable $A$, the probability of measuring a value between $a$ and $a+da$ is given by $|\langle a|\psi \rangle |^2da$ (and furthermore, if $a$ is not an eigenvalue of $A$, then the probability of measuring a value in this interval is $0$). How would you convince your students that this had to be the case? I have thought about this question of motivation for a couple of years now, and so far, the only answers I've come up with are incomplete, not entirely satisfactory, and seem to be much more non-trivial than I feel they should be. So, what do you guys think? Can you motivate the usual formulation of quantum mechanics using only classical mechanics and minimal appeal to experimental results? Note that, at some point, you will have to make reference to experiment. After all, this is the reason why we needed to develop quantum mechanics. In principle, we could just say "The Born Rule is true because its experimentally verified.", but I find this particularly unsatisfying. I think we can do better. Thus, I would ask that when you do invoke the results of an experiment, you do so to only justify fundamental truths, by which I mean something that can not itself just be explained in terms of more theory. You might say that my conjecture is that the Born Rule is not a fundamental truth in this sense, but can instead be explained by more fundamental theory, which itself is justified via experiment. **Edit** : To clarify, I will try to make use of a much simpler example. In an ideal gas, if you fix the volume, then the temperature is proportional to pressure. So we may ask "Why?". You could say "Well, because experiment.", or alternatively you could say "It is a trivial corollary of the ideal gas law.". If you choose the latter, you can then ask why that is true. Once again, you can just say "Because experiment." or you could try to prove it using more fundamental physical truths (using the kinetic theory of gases, for example). The objective, then, is to come up with the most fundamental physical truths, prove everything else we know in terms of those, and then verify the fundamental physical truths via experiment. And in this particular case, the objective is to do this with quantum mechanics.


Why does the addition of paramagnetic ions decrease the spin lattice relaxation time of protons?

My question pertains to NMR (nuclear magnetic resonance). $T_1$, the spin lattice relaxation time of protons in water is about 2.5 seconds. If you add some $CuSO_4$ (copper sulfate) to the water the spin lattice relaxation decreases dramatically. I.e. in a 1mM $CuSO_4$ solution, $T_1$ drops to about 0.46 seconds. Why is this? I've read that paramagnetic ions decrease spin lattice relaxation times, but how exactly does this work?


Common approximations for fluid outflow beyond Torricelli's law

Last month, a puzzle here in physics.stackexchange asked to calculate the dynamics of the outflow in a loose (over rails, no friction) tank using Torricelli's law as a simplification. The simplification turned out to be problematic; one would prefer some solution where the height $h(0)=h_0$ starts decreasing with the usual condition $h'(0)=0$. A first try to generalize is to use the potential energy stored in the tank to provide kinetic energy both to the CoM of the fluid and to the outflow. Consider for instance two horizontal equal nozzles, so that the CoM only goes down vertically. While this situation drives to a manageable second order differential equation, I am not sure if it is good enough. Particularly it seems that when we close the nozzle(s), so that the CM stops, this kinetic energy should change to be stored as internal energy, does it? We could imagine some configuration where we open the nozzle an interval $\Delta t$, close it, open it at the next interval, &c, so that we force this conversion to internal, not CoM, energy. So, is there some generic condition where we can grant that all the energy goes either to the outflow or to the fluid CoM? If not, there is some usual parameter measuring how much of the energy goes to CoM and how much goes to internal (vorticity or so)?


What is a maximal analytic extension?

Can someone explain (as rigorously as possible) what is involved in analytically continuing, say, the Schwarzschild solution to the Kruskal manifold? I understand the two metrics separately but I'm not sure how analytic continuation is used, since I can't really see how the process of extending the domain of a complex function has anything to do with extending a manifold through a coordinate change.


Interpretation of QM as discrete derivative

In a couple of letters dated 1923, Pauli writes to Sommerfeld and Lande about Zeeman effect, and he describes (a piece of) quantisation in a peculiar way: the substitution of the " _Differentialquotient_ " $$ {d \over dj} {1 \over j} $$ ($j$ is called a _impulsquantenzahlen_? is it an angular momentum, isn't it?) by the " _Differenzen quotient_ " $$\frac 1j - \frac 1{j-1}$$ (see page 3 of letter to Sommerfeld). The later being argued as plausible from the integration of $\int {dj \over j^2}$ (footnote in page 4 of letter to Landé). I wonder, did the concept of " _Differenzen quotient_ " survive in the literature, as a quantisation method or even as some fundational concept of quantum mechanics?


Why does Quantum Electrodynamics Allow a Photon to Exist Temporarily as a Positron and an Electron?

In this question... Why does a photon colliding with an atomic nucleus cause pair production? ...I asked why a photon colliding with a atomic nucleus can become an electron and a positron. The answer that I thought was most illuminating explained that a photon spends some of its travel time as a particle-antiparticle pair of an electron and a positron. If it strikes the nucleus at the right time, this pair will be separated. It was explained that this is because 'Quantum Electrodynamics allows it'. Why does Quantum Electrodynamics allow a photon to exist temporarily as an electron and a positron?


Orthogonality of the wavefunctions on an subinterval?

Lets say that functions (eigenfunctions) $\psi_0$ and $\psi_1$ are orthogonal on an interval $-d/2 < x < d/2$. Are they also orthogonal on any subinterval inside the interval $-d/2 < x < d/2$? For example if I take subinterval $$d/4 < x < d/2$$ Can I state that: $$ \int\limits_{d/4}^{d/2}\overline{\psi}_0\psi_1\, dx= 0 $$ This would easier a lot of integrations for me...


Very large absorption lines in stellar spectrum

I was puzzled by the wide absorption lines in a stellar spectrum I found. The following is what I expect absorption lines to look like - thin, crisp lines: ![enter image description here](http://i.stack.imgur.com/RMHmB.gif) However, I found this stellar spectrum, which as large gaps, most notably from 6600 to 7100 angstroms: ![enter image description here](http://i.stack.imgur.com/97J5O.gif) Why are the absorption lines so wide on the second spectrum? Are they even absorption lines at all, or another phenomenon altogether?


Calculating the mass equivalency of a song?

I've recently become fascinated with the idea of sound energy having a theoretical equivalent mass. I've read over this thread: Do light and sound waves have mass I understand this part: $m_{eq}=E/c^2$ and $E=A\rho \xi^2\omega^2$ Where I am getting tripped up is the way to measure $E$. Essentially, I want to measure $E$ and eventually $m$ of a song (for it's entire duration). The only equipment I have is an iPhone app that measures intensity (dB) (apparently it's fairly accurate for levels below 100 dB). I plan to play the "music" at about 80dB over speakers in a room that is 50'x30'x15' and I will assume the temperature is at room temperature. If your wondering, it's for a conceptual audio art piece.


Are two waves being in phase is the same as saying that the two waves are coherent?

If two waves are coherent, is it the same as them being in phase? Please correct if I'm wrong.


Is the law of conservation of energy still valid?

Is the law of conservation of energy still valid or have there been experiments showing that energy could be created or lost?


motion in the body-fixed frame?

This is really basic, I'm sure: For rigid body motion, Euler's equations refer to $L_i$ and $\omega_i$ as measured in the fixed-body frame. But that frame is just that: fixed in the body. So how could such an observer ever measure non- zero $L$ or $\omega$?


Is ghost-number a physical reality/observable?

One perspective is to say that one introduced the ghost fields into the Lagrangian to be able to write the gauge transformation determinant as a path- integral. Hence I was tempted to think of them as just some auxiliary variables introduced into the theory to make things manageable. But then one observes that having introduced them there is now an extra global $U(1)$ symmetry - the "ghost number"   * Hence hasn't one now basically added a new factor of $U(1)$ to the symmetry group of the theory? How can the symmetry of the theory depend on introduction of some auxiliary fields?    * Now if one takes the point of view that the global symmetry has been enhanced then the particles should also lie in the irreducible representations of this new factor. Hence ghost number should be like a new quantum number for the particles and which has to be conserved!    * But one sees that ghost field excitations are BRST exact and hence unphysical since they are $0$ in the BRST cohomology.  I am unable to conceptually reconcile the above three ideas - the first two seem to tell me that the ghost-number is a very physical thing but the last one seems to tell me that it is unphysical.   * At the risk of sounding more naive - if the particles are now charged under the ghost number symmetry then shouldn't one be able to measure that in the laboratory?    * Lastly this ghost number symmetry is a global/rigid $U(1)$ symmetry - can't there be a case where it is local and needs to be gauged?


Is the crystallization of a heat pack the same as the crystallization of freezing water? If so, why does one generate heat where the other does not?


What does ionization of neutral Hydrogen have to do with "transparency"?

Most accounts of the early history of the Universe make some reference to (re)ionization as being the _reason_ that the Universe becomes transparent after a period of opacity caused by the absence of ionized hydrogen. These accounts suggest that neutral H is opaque, wile ionized H is transparent. But I don't follow the reasoning. I understand that, while neutral H is primarily absorbing at a single relevant wavelength, those wavelengths are "smeared out" by expansion, as seen by an observer, so that a dense and nearly continuous neutral H at a large range of redshifts would result in absorption at a corresponding wide range of wavelengths (i.e. "opacity" at those wavelengths, as seen by an observer "on the other side" of the absorbing intervening H). But I'm not clear on why ionization of the same H, alone, would result in clearing up this opacity.


Molecular field meaning in Liquid Crystal Theory

Given the Frank-De Gennes free energy $F = \int f(\boldsymbol{p},\nabla\boldsymbol{p}, ...)\ d\boldsymbol{x},$ for liquid crystals (see De Gennes-Prost, p. 107, formula 3.21), the vector $$h_{i}=-\frac{\delta F}{\delta P_{i}}=-\frac{\partial f}{\partial P_{i}}+\partial_{j}\frac{\partial f}{\partial\partial_{j}P_{i}}$$ is usually called _molecular field_ , a "notation derived from magnetism". What is the physical meaning of this vector in this context? Does it have any sense to call it _molecular_ also here?


A simple question on the projected wave function?

For example, consider a spin-1/2 AFM Heisenberg Hamiltonian $H=\sum_{<ij>}\mathbf{S}_i\cdot\mathbf{S}_j$, and we perform a Schwinger- fermion($\mathbf{S}_i=\frac{1}{2}f^\dagger_i\mathbf{\sigma}f_i$) mean-field study. Let $H_{MF}=\sum_{<ij>}(f^\dagger_i\chi_{ij}f_j+f^\dagger_i\eta_{ij}f_j^\dagger+H.c.)$ be the resulting mean-field Hamiltonian, where $(\chi_{ij},\eta_{ij})$ is the mean-field ansatz. And let $\psi_{1,2}$ represent two exact eigenstates of $H_{MF}$ with energies $E_{1,2}(E_1>E_2)$, say $H_{MF}\psi_{1,2}=E_{1,2}\psi_{1,2}$. Now we can construct the physical spin states $\phi_{1,2}$ by applying the projective operator $P=\prod_i(2\hat{n}_i-\hat{n}_i^2)$(Note that $P\neq \prod _i(1-\hat{n}_{i\uparrow}\hat{n}_{i\downarrow})$) to $\psi_{1,2}$(where $\hat{n}_i=f^\dagger_{i\uparrow}f_{i\uparrow}+f^\dagger_{i\downarrow}f_{i\downarrow}$), say $\phi_{1,2}=P\psi_{1,2}$, and _generally we don't expect that $\phi_{1,2}$ are the exact eigenstates of the original spin Hamiltonian_ $H$. My question is: $\frac{\left \langle \phi_1 \mid H \mid \phi_1 \right \rangle}{\left \langle \phi_1 \mid \phi_1 \right \rangle}>\frac{\left \langle \phi_2 \mid H \mid \phi_2 \right \rangle}{\left \langle \phi_2 \mid \phi_2 \right \rangle} $ ? If it's true, then how to prove it rigorously ? Thanks a lot.


Free Will Theorem question

The Kochen-Specker Theorem says, if I understand it correctly, that the results of spin measurements cannot be predetermined independent of measurement. They get to this conclusion by describing 33 possible measurements whose results cannot all be satisfied simultaneously. In the Free Will Theorem, this is developed further by taking two entangled particles sufficiently far away and measuring in 2 of these 33 directions. The proof seems to assume consistency of these results, so my question is this: If we had an entangled group of 33 particles rather than just 2, what would happen if all 33 directions were measured by space-like separated experimenters? The Kochen-Specker Theorem says that results can't all be consistent with one another. I have very little background in physics so I apologize in advance if I'm misunderstanding something fundamental. I would also request that answers assume little knowledge of physics beyond what I've displayed above, though knowledge of mathematics can be assumed.


Batteries and voltage?

The _voltage_ of a battery gives you the difference in potential energy 1C of charge would have at the positive terminal vs the negative terminal. If I connect a wire to both terminals, the battery will generate a field within the wire. The field will cause surface charges to buildup on the wire. You essentially end up with surface charges distributed through the wire so that the field is always parallel to the surface of the wire. Now say I have an unconnected battery of 5V. Now hypothetically if I used "tweezers" to drag 1 coloumb of charge through a vacuum from the positive terminal to the negative terminal it would gain 5J of energy. Now consider a connected battery. If I used "tweezers" to drag 1 coloumb of charge through the _wire_ from the positive terminal to the negative terminal it would _still_ gain 5J of energy. My question is that since the surface charge in the wire sets up a field, how do we know that the field set up by the surface charge doesn't affect the potential energy difference between the positive and negative terminal? Voltage is defined as the line integral of the dot product of the electric field and distance, and since the electric field acting is not longer the same (since the surface charge also contributes a field) how do we know that 1C of charge would gain the same amount of energy if dragged from the positive terminal to the negative terminal? ![enter image description here](http://i.stack.imgur.com/5nFRk.png)


The "official" mass of the sun as unit?

I'm searching for the "official" mass of the sun as a unit in astrophysics. The mass of the sun can be calculated by: $M_{\odot}=\frac{4\pi^2\times(1 \ \text{ua})^3}{G\times(1\ \text{year})^2}$ * * * So in this formula: $\pi = 3.1415926535898...$ $1 \ \text{ua} = 149597870700 \ \text{m}$ (it's an exact definition according to the IAU 2012 resolution http://www.iau.org/static/resolutions/IAU2012_English.pdf) $G = 6.67384\times10^{-11} \ \text{m}^3\text{kg}^{-1}\text{s}^{-2}$ according to CODATA (http://physics.nist.gov/cgi-bin/cuu/Value?bg) but $G = 6.67428\times10^{-11} \ \text{m}^3\text{kg}^{-1}\text{s}^{-2}$ according to IAU 2009 (http://maia.usno.navy.mil/NSFA/IAU2009_consts.html), so which one do we choose? Finally, for the year here, which year to choose? Is it the same year as for the light speed: $1 \ \text{year}=365.25\times24\times3600=31557600 \ \text{s}$ (Julian year)? Or the tropical year $1 \ \text{year}=365.2421897\times24\times3600=31556925.2 \ \text{s}$ (tropical year)? * * * For some of the preceding values we obtain: $M_{\odot}=\frac{4\pi^2\times(149597870700)^3}{6.67384\times10^{-11}\times(31557600)^2}=1.988622... \ \text{kg}$ which is not what gives wikipedia (http://en.wikipedia.org/wiki/Solar_mass) or google (https://www.google.fr/search?q=solar+mass) So my question is: what are the correct preceding values, and consequently what is the current good value for the sun mass? (as some units may have been redefined since some measurements it's not that simple)


Do small-scale physical effects lead to larger systems being required for long life (of dynamic systems, not static objects)

I had a hard time finding a tag, but I think physicists can deal with this question best. I answered a question about "immortality" when some guy claimed I got it wrong that neurons die (I argued that even if you live a billion years you die slowly many times over because all your cells incl. nervous cells will have to be replaced, meaning you will regularly - but never suddenly - become a new person; also: continuous learning, and information storage is physical, and our brain has very limited capacity so memories will change), immortal means they are immortal. That got me thinking: Is it actually possible to have repair systems in each cell, or isn't it better to have a repair system on a much higher macro level and cell-level immortality is actually impossible? Background: The smaller the scale, the more events like quantum effects and Boltzman (energy) distribution (e.g. random atom movements, crystal structure defects, breaking bonds) dominate. Meaning small systems WILL break in unpredictable ways, so only large systems can live long and have reliable repair systems because the larger the system the less important those effects become. Summary: It is much better (or even possible) to have big long-living systems than small ones, because the inevitable repair system cannot be too small because it will be subject to random events of the micro-world. An "immortal" human would have a body-level repair system that repairs by letting broken cells die (we have that already) and build a new one, instead of having a repair system on the cell-level that would make cells immortal (too unreliable, won't work). Or in other words: The smaller the system the less likely a long life can be achieved. Note that I'm asking about complex dynamic systems, not static objects or information. Like living organisms or (complicated) machines, and about their chance to "live long".


Is data which rides on the carrier frequency dangerous?

My understanding of broadcasting data via electromagnetic radition is, that the data "rides" on a carrier frequency on which it is radiated. I am aware of the fact, that those carrier frequencies are below the frequencies of visible light, thus not dangerous like the higher ones (UV, Xray, Gamma). My question now is, if the data (be it an analog tune from a radio, or a digital WiFi signal) that "rides" on this carrier frequency is oscillating fast enough that it can be considered harmful, because it belongs to the high frequencies greater than visible light.


Deriving group velocity formula

A formula for the group velocity of waves is: $u=k*dv/dk + v$ But then, since $k=2π/λ$, this equation can be rewritten as: $u=v-dv/dλ*λ$ But how? My attempt: $k*dv/dk$ = $((2π/λ)*dv/d(2π/λ)$ Then what?


What information is lost in the symmetrization necessary to derive the BBGKY hierarchy?

The book on Kinetic theory I'm reading derives the BBGKY hierarchy after introducing the reduced distribution functions $f_s(q^1,p_1,q^2,p_2,\dots,q^s,p_s):=\int\ \rho\ \ \mathrm d q^{s+1} \mathrm d p_{s+1}\cdots \mathrm d q^N \mathrm d p_N,$ where $\rho$ is the phase space probability function. In fact, in the derivation, to get rid of most terms, he reasonably states that we must make the assumption that the $f_s$ are symmetric, i.e. $\forall k.\ f_s(\dots,q^k,p_k,q^{k+1},p_{k+1},\dots)=f_s(\dots,q^{k+1},p_{k+1},q^k,p_k,\dots).$ The step from going from the particle trajectory $\pi$ (which solves the Hamiltonian equations of motions) to the distribution $\rho$ (which solves the continouty equation with the vector field, generated by multiple $\pi$-streamlines, as flow) can be a bit confusing. The fact that the author concerns himself two times with this and doesn't just say "we can clearly assume that.." suggests to me that it might not be so simple to put asside. He offers a symmetrization method $f_s\mapsto \frac{1}{s!}\sum_{\pi}f_s$, where the sum permutes the arguments. Now firstly, I wonder how this works with multi-particle distribution functions. What do the equations look like for an $f_{s,t}$ which describe $s$ particle of one sort and $t$ particles of another sort? Also, as this implies even the Boltzmann equation rests on this assumptions which therefore apparently isn't initially part of the Hamiltonian description - what information is lost in the process of symmetrization?


Reflections in a double-glazed window

Recently I observed reflections in a double-glazed window. I wondered that the double reflections were not the same if I changed my point of view. For example the two images of a candle would overlap at a certain angle but separated when I moved. The only idea I had to explain this behavior is that one (or both) of the glass panels was not totally plane but curved. Does that sound plausible or is there another explanation for this behavior?


Sudden Approximation for Beta Decay of Tritium Atom

I am working out this problem right now, and I'm confused by the answers I'm getting. Problem: A tritium nucleus (Z = 1) in a tritium atom undergoes beta decay, i.e., a neutron in the nucleus emits an electron and transforms into a proton (Z = 2). Assuming that the emitted electron acquires so much energy that becomes unbound and flies away almost instantly without interacting with the orbiting electron, (i) Compute the probability that the bound electron will be in a state with principal quantum number n in the Helium ion. Assume that initially the bound electron was in the ground state of the tritium atom. Attempt: The new wavefunctions differ by Z=2, and we want to calculate transition probability between the old ground state and the new wavefunctions, so we take the inner product between them. Looking at the hydrogen wave function, by orthogonality of the spherical harmonics the inner product will vanish if the l's aren't equal, so l'=l=0, which simplifies our inner product, which is now an integral over r. My final expression for the transition probablity is: (Integrate[(8*Sqrt[2]/a^3)*1^(-5/2)*LaguerreL[1 - 1, 1, 2*2*r/(1*a)]* LaguerreL[0, 1, 2*r/(a)]*Exp[-r/a - 4*r/(1*a)]*r^2, {r, 0, Infinity}])^2 or 128*(Integrate[ n^(-5/2)*LaguerreL[n - 1, 1, 4*r/n]*Exp[-r - 4*r/(n)]*r^2, {r, 0, Infinity}])^2 So I get an expression of 512/15625 for the probability it is in the new ground state. This is really low. I am supposed to calculate the infinite sum of these probabilities as well, with mathematica, but it can't figure it out. So either I'm doing something wrong in my thought process, or there is a way of writing of the laguerre polynomial that simplifies our problem considerably.


For which irreps can normalized spin vectors be chosen continuously, for all possible spins?

Let $(u_x,u_y,u_z)$ be a unit vector. Then a normalized vector for spin-1/2 in this direction is given by normalizing a column from the pure density matrix: $\frac{1}{\sqrt{2+2u_z}}\left(\begin{array}{c}1+u_z\\\u_x+iu_y\end{array}\right)$. This solution works except when $u_z=-1$. So this gives a normalization choice of spin vector that is continuous over the possible eigenstates of spin with the exception of spin down = $\left(\begin{array}{c}0\\\1\end{array}\right)$. It is impossible to choose normalized spin-1/2 vectors for all possible eigenstates of spin-1/2. This is a fact that is related to Berry-Pancharatnam phase. The best you can do is to leave a single point, as in the above, where there is an essential discontinuity. What I'd like to know is what other j have spin-j vectors unable to be normalized. That is, given all possible eigenstates of spin-j, is it possible to choose one from each ray, such that the choice is continuous? * * * Partial Solution: Per Dr. Motl's answer below, I'll down the situation for a natural representation of spin-1. Choose $S_x = i\left(\begin{array}{ccc}0&0&0\\\0&0&+1\\\0&-1&0\end{array}\right)$ $S_y = i\left(\begin{array}{ccc}0&0&-1\\\0&0&0\\\\+1&0&0\end{array}\right)$ $S_z = i\left(\begin{array}{ccc}0&+1&0\\\\-1&0&0\\\0&0&0\end{array}\right)$. It's easy to verify that these satisfy the $SU(2)$ rules $[S_x,S_y] = iS_z$, etc. Then the spin operator for spin in the $u = (u_x,u_y,u_z)$ direction is given by: $S_u = i\left(\begin{array}{ccc}0&u_z&-u_y\\\\-u_z&0&u_x\\\u_y&-u_x&0\end{array}\right)$ and the solution with spin=0 in the $u$ direction is given by the vector $u$. And sure enough, it's impossible to choose the vectors for spin=1 as a continuous function of $u$.


Should the cold drink come out of a free falling bottle with an open lid?

If I hold an open bottle and perform free fall, say sky diving, along with it from, say 10000ft, while I'm holding the bottle vertically upwards should the cold drink come out of the bottle? Kindly ask for any further details you need.


When does a function of an operator act in the same way as the operator?

"Consider an operator $A = r - a$, where $r$ is an operator and $a$ is a constant. Consider only those state kets $V_i$ in the Hilbert space such that $AV_i = 0$ ($A$ acting on $V_i$). Define a projection operator $P_A$ that projects an arbitrary ket on to the states $V_i$'s, i.e. the subspace of kets annihilated by $A$. Then $P_AAP_A = 0$. So $P_ArP_A = aP_A$." Now further in the literature which includes whatever I've quoted above, the following is stated: Suppose $H = (1/r^2)p_\theta^2 + \lambda(r - a)$ (where $p_\theta$ is an operator). It follows that $P_AHP_A = (1/a^2)p_\theta^2$. Could anyone please elaborate how the last implication comes about? Is it that since $rV_i = aV_i$, $(1/r^2)V_i = (1/a^2)V_i$? Thanks...


Quantization of Gravitational Field: Quantization conditions

I'm begining to study Quantization of field with the second quantization formalism. I've studied phononic field, electromagnetic field in the vacuum and a generic relativistical scalar field. I asked to me if is possible doing the same thing with the Gravitational Field Hamiltonian. I've heard that we can do it only in the condition of linearized Gravity and we obtain a field with spin 2, but we can't do it in the general condition because we don't have quantization conditions. What are these conditions? And how can we obtain only in linearized gravity the quantized field? And how it has got spin 2?


As ISA Practical - Resistors in Parallel

> **Possible Duplicate:**   >  Current against the inverse of resistance graph, I = V/R +c How would you set up a circuit with a fixed resistor in parallel with a variable one. We are told to measure the current across the combination resistors and our Voltage is $1V$ and then construct an $I$ against $1/R$ graph what sort of correlation would this be?


Why does a window layer is on top of the absorber layer in solar cells and not below it?

In solar cells there is a p-n junction. P-type semiconductor (for example CdTe) is often absorber layer because of its carrier lifetime and mobilities. In case of CdS/CdTe, CdS is n-type window layer and everywhere it is said that it should be very thin and has large band gap – not to absorb any light and let it go through to the p-type absorber (that is why it is called a window layer). But why should it be on top of the absorber layer and not below it? If n-type layer is below, the light can hit the p-type absorber directly. I have some ideas that it is related to the distance between the place of absorption and p-n junction, but I am not sure. ![CdTe solar cell](http://i.stack.imgur.com/8IajS.jpg) Image by Alfred Hicks/NREL


How can scintillation gamma-spectrometers work given that track length is different for different angles?

As far as I understand, the basic principle of gamma spectrometer is simple - gamma ray hits scintillator, it generates number of photons which roughly proportional to gamma ray energy. Then we need to detect these secondary photons and plot a nice graph. But as far as I see, scintillator size is typically too small to completely absorb gamma photon. Which also means that gamma photons coming from different angles will have different track lengths inside scintillator - and should generate different amount of light for the same gamma ray energy. So, how can this work at all?


Reference request for the Aharonov-Bohm effect

I am looking for a good reference to an online source or book, on the magnetic Aharonov-Bohm effect. I have read the appropriate sections from the book by Griffiths and Ballentine, and still haven't understood it properly. The reference should also contain topological arguments. I have done QM on the order of 4 chapters of Griffiths, and the first 2.5 chapters of Sakurai.


Kinematic sign convention

For example, if I drop a ball from a $50$ meters building, then I will consider   1. the ground is $0$ meter   2. downward is positive ( which makes gravity positive, downward velocity positive, etc) so with that if i use $X_f = X_i + V_it + \frac{1}{2}at^2$ then I would get something like $0 = 50 + 4.9t^2$ which is not even possible. Instinctively I know what to do but when I think more about sign convention, it seems so confusing.


Deriving $p = mv$ from translational symmetry (momentum conservation law)?

> "In classical mechanics, momentum is defined as the quantity which is > conserved under global spatial translations or, alternatively, as the > generator of spatial translations." (G.Parisi, Quantum mechanics) From this definition (the generator of spatial translation), how can I obtain the analytical expression of the momentum (i.d. $p=mv$)?


Is the covariance or contravariance of vectors/tensors something that can be "visualized"?

I'm taking an undergrad GR course, and our text (Lambourne) mentions covariant and contravariant vectors and tensors ad-nauseum, but never really gives a formal definition for what they are, and how they are unique from each other in any physical sense (other than their difference in transformations). Is there any physical intuition behind these two labels? There should be, right? If they differ in how they transform with transformation of coordinates, doesnt that indicate that there has to be some way of visualizing their difference, since coordinate transformations are easily visualized?


Question about liquid / gaseous insulators

I'm looking for information about liquid / gaseous insulators, but I can't find any, is there any website (because I can't find on Wikipedia) where I can get this information? Also, can I get a list of examples of liquid / gaseous insulators? Any kind of help is appreciated.


Where can I find simulation software for electricity and magnets?

Is there easily-available* software to simulate coils, solenoids, and other magnetic and electromagnetic devices? I'd like to play around with some design ideas, such as Halbach arrays, but physics simulators like SketchyPhysics don't seem to have the laws of electromagnetism implemented. (*"Easily-available" means "not cost prohibitive.")


If a magnetic monopole falls into a schwarzchild black hole, what happens to the magnetic field?

By the no-hair theorem, black holes can only have mass, charge and angular momentum. Does "charge" include "magnetic charge" (such as from a magnetic monopole)? Can black holes have magnetic charge in addition to electric charge? [By the way, I believe theory says magnetic monopoles have to exist, the reason why we haven't discovered any is because inflation diluted them to such an extent that there are none or only a few in our Hubble volume.] * * * These three questions are similar but I think the answers will be different for each one: > What happens to an embedded magnetic field when a black hole is formed from > rotating charged dust? It seems to me a rotating charged black hole must > have a dipole magnetic field. But the strength of the dipole field seems > like an extra parameter that black holes are forbidden by the no-hair > theorem. > > If a magnetic monopole falls into a schwarzchild black hole, what happens to > the magnetic field? Here there would be only radial magnetic field lines > leaving from the event horizon to infinity. So if magnetic charge is counted > as charge this should be no problem. But if the black hole were rotating > wouldn't that produce an electric dipole field? > > When a neutral star with a magnetic field collapses to form a black hole, > what happens to the magnetic field? Here there is no charge so how can there > be a magnetic field associated with a black hole? That would definitely > violate the no-hair theorem.


Differences the nonlinerarties

I want to comparison between oscillons based on non-linearities. Can someone elaborate it with the reason behind it : > When the sinusoidal vibrations are of the correct amplitude and frequency > and the layer of sufficient thickness, a localized wave, referred to as an > oscillon. > > Astonishingly, oscillons of opposite phase will attract over short distances > and form 'bonded' pairs. Oscillons of like phase repel. Oscillons have been > observed forming 'molecule' like structures and long chains. In comparison, > solitons do not form bound states. Reference: Wiki


Hydrogen 2P Probability Density Question

I'd like to calculate the probability density for Hydrogen in the $|2,1,1\rangle$ and $|2,1,-1\rangle$ states. There is an $\exp(\pm i\phi)$ term attached to the wave function for these states. It seems to me that $|\Psi|^2$ in this case would be the same for $|2,1,1\rangle$ and $|2,1,-1\rangle$ since $|\exp(i\phi)| = 1$ with the same being true for $|\exp(-i\phi)|$. Where am I going wrong here? Also, since I'm working in spherical coordinates the PD should be $r^2|\Psi|^2$, right (this will result in those classic energy orbital diagram functions)?


Effective mass of a black hole?

Suppose a black hole forms from a given mass of particles such as the core of a star going supernova. The black hole formed will have an effective mass due to the curvature of space time induced. Such a mass is presumably the inferred mass deduced from the effect of the black hole on the motion of nearby bodies. How does this effective mass compare with the original mass of particles all of which ended up "in the black hole"? Is the effective mass less greater or the same as the formation mass?


I want to charge 48V 108Ah battery from 48V 80Watt solar panel. Can anyone write that how much time will take this battery to fully charge?

> **Possible Duplicate:**   >  Charging 12V 150Ah battery I want to charge 48V 108Ah battery from 48V 80 Watt solar panel. Can anyone write that how much time will take this battery to fully charge?


Molecular formula of a compund

I have no idea on how to get information out of this one. > A compound contains only hydrogen and nitrogen. Nitrogen makes up for 87.4% > of the mass. A sample of this have a density of 0.977 g/L at 740 torr and > 100 C. What is the molecular formula of the compound? The best I could think of was to put it in $PV=nRT$, so $n=PV/RT$. But since there is no defined volume I can't put in anything for $V$.


Can acceleration be achieved without reaction mass?

In a previous question, I asked whether a flashlight can accelerate me. The answer is yes. This has led me to wonder, this is inferior to all other known propulsion methods in all aspects except one: Shooting light (photons) accelerates the shooter without ejecting any mass. This means we can space travel forever, assuming we have enough electricity. **Are there any other ways to accelerate without losing mass?** Note that there are lots of way to do it with external forces like laser propulsion, but I am only interested in doing it without those. e.g. consider a spacecraft swimming in the void trying to accelerate.


Why do excitonic absorptions have small bandwidth?

Below is an image of the optical density (proportional to the absorption coefficient) of KBr crystal at low temperature. Indicated at 6.6 ev and 7.7 eV are the absorption by excitons. As you can see, it has a small width. Can you explain why? ![enter image description here](http://i.stack.imgur.com/NQgRN.jpg) Source:Theory of Excitons, Robert S Knox, 1954


How must you spin the ball to make it alternate between 2 positions?

![enter image description here](http://i.stack.imgur.com/y6JZ2.jpg) Assume any parameters you may need. Thanks in advance.


What is the limit of vanishing density?

I've been asked to examine the limit of vanishing density for the virial equation of state derivation of $\left(\frac{\partial U}{\partial V}\right)_T$ and quite frankly, I have no idea what this is and a Google search doesn't seem to net anything extremely helpful. Assuming density can be isolated in my derivation above, am I just taking the limit as $\rho \rightarrow 0$ ?


Can a black hole bounce?

Is there a limit to the amount of matter that a black hole can accrete per second and if so could a certain sized black hole bounce off a dense enough surface?


Questions about the speed of light and the extent of the universe

What is the reason that speed of light is the maximum speed in the universe? Why there is no speed swifter than light? Or is it just we haven't found yet? A similar question is like this: why the light speed in vacuum has to be 299,792,458 m/s? Why couldn't it be faster? Who or what is controlling this speed in vacuum? Who is stopping it from being faster? Another question confusing me is this, if light from a far away Galaxy travels to us and we see the stars tonight, couldn't it be fully possible that at this time, they actually are already dead? Which means by modern scientific methods, there is absolutely no way to know what is really happening out there, all we know is just what was there, right? So our knowledge is going to be forever limited unless we have something that can travel much faster than light?


Relation between electric charge and gauge parameter of the moduli space of monopoles

I am studying about the moduli space of a 2 monopole system from Harvey's notes, and Manton's paper. In both of these, (Harvey section 6.2), after constructing the Lagrangian for a two dyons system, the author replaces the electric charge $e$ with $\dot{\chi}$. $$e \to \dot{\chi}$$ to obtain the the equation of motion is geodesic form, from which the Taub-NUT metric can be seen. Why is this identification made? Why does the rate of change of the gauge parameter, give the electric charge?


A James Clerk Maxwell Disproof

One of my favorite physicists to learn about was James Clerk Maxwell, for the fact that he unified the study of E&M in physics and he would often disprove theories that did not work as a Mathematician. I remember my physics professor mentioning a disproof Maxwell had done with what was then the current knowledge of how currents flowed or were held through capacitors... This is probably way off from what he actually did disprove since it has been over 5 years since I had been to this particular lecture but could anyone tell me or demonstrate the exact disproof by counterexample that Maxwell had done with capacitance?


How do you super impose two or more signals to occupy a fix area of space with the resultant summed wave?

Is it possible to super-impose two or more signals all sent from different directions as a standing wave with the resulting summed wave occupying a fix area of space that is also a complex area? Do you understand my question? It is not obvious probably what i am trying to say-feel free to ask for more of an explanation. These signals could be electro magnetic waves or micro waves designed to pass thru solid objects under normal circumstances but then excite the solid only when at a certain frequency - this excite frequency would come about when two or more signals interact within the solid.


In dimensional analysis, why the dimensionless constant is usually of order 1?

Usually in all discussions and arguments of scaling or solving problems using dimensional analysis, the dimensionless constant is indeterminate but it is usually assumed that it is of order 1.   1. What does "of order 1" mean? 0.1-10?   2. Is there any way, qualitative or quantitative, to see why the dimensionless constant is of order 1?   3. Are there exceptions to that? I mean cases where the dimensionless constant is very far from 1? Could you give some examples? Can such exceptions be figured out from dimensional analysis alone?


What is the physical significance of dot & cross product of vectors? Why is division not defined for vectors?

I get the physical significance of vector addition & subtraction. But I don't understand what do dot & cross products mean? More specifically,   * why is it that dot product of vectors $\vec{A}$ and $\vec{B}$ is defined as $AB \cos{\theta}$   * why is it that cross product of vectors $\vec{A}$ and $\vec{B}$ is defined as $AB \sin{\theta}$ To me, both these formulae seem to be arbitrarily defined (Although, I know that it definitely wouldn't be the case). If Cross product could be defined arbitrarily, why can't we define division of vectors? What's wrong with that? Why can't vectors be devided?


> In thermodynamics, the Helmholtz free energy is deemed as a thermodynamic > potential which calculates the “useful” work retrievable from a closed > thermodynamic system at a constant temperature and volume. For such a > system, the negative of the difference in the Helmholtz energy is equal to > the maximum amount of work extractable from a thermodynamic process in which > both temperature and volume are kept constant. In these conditions, it is > minimized and held constant at equilibrium. (citation) My questions are   1. How can one say that if we subtract "TS" from "U" then what we get is free energy? Why do we have to subtract "TS"? What if we subtract pV instead of "TS".   2. If the system is in constant temperature and Volume , then how it is possible to do work?   3. Is it the statement true for Ideal gas also. If it is true then a thermodynamic system(with ideal gas) at a constant temperature and volume is also at constant pressure. so it is at equilibrium. Does that mean for a ideal gas system free energy is always minimum?


Dependence of Friction on Area

Is friction really independent of area? The friction force, $f_s = \mu_s N$. The equation says that friction only depends on the normal force, which is $ N = W = mg$, and nature of sliding surface, due to $\mu_S$. Now, less inflated tires experiences more friction compared to well inflated tire. Can someone give clear explanation, why friction does not depend on area, as the textbooks says?


Friction force and contact area

It's usually stated that the friction force is independent of the area of contact (Amontons' Second Law). I've always thought that this shouldn't be true, because the atraction between molecules would be higher and there would be more peaks. I've read that this is important for rubber surfaces. Is it important for nonelastic materials, like wood? Are there any quantitave studies of this phenomenon?


De-excitation and vibrational frequency

By utilising energetic beams of particles it is possible to set atomic nuclei into vibration. For a certain nucleus the subsequent de-excitation results in the emission of photons of energy $1$ $MeV$. Use this information to make an estimate of the vibrational frequency of the nucleus, stating any assumptions that you make. Using $E=hf$ I can calculate the frequency of the energetic beams. However I don't get the vibrational frequency bit...


What is the difference between scale-free network and small-world network? I can't understand from the definitions around the web if they are both the same name for one thing. Do both follow a power-law distribution of the nodes degree? In particular is it true that scale-free networks are a subset of small-world networks?


What forms are theoretically predicted for orbitals or quarks in hadrons and of hadrons in tritium?

We all know very illustrative spatial representations of predicted electron orbitals in atoms which are essentially spatial plots of the solutions of wave equations. In all atoms the electrons occupy the positions around the nucleus, sometimes, spatially symmetric. But I wonder which form could take orbitals of three particles of nearly- similar mass, such as quarks in a hadron or hadrons in tritium nucleus? Seems there is no place in the center for one of them...


Objects falling on Slopes

An object falls on a slope and then rebounces....and it is known it hits the slope again...how do I calculate it's second point of contact with the slope....how can a projectory equation be used in this case?


Making elementary guess regarding current in AC circuits

I was just studying the comparisons, between A.C, and D.C, and the effects they can produce in electrical circuits (viz, Joule-heating, Peltier-effect and Seebeck-effect in thermocouples, Electrolysis, Magnetic effect). I found from what is written in my textbook, that all of these effects could be produced by a purely D.C circuit, but a purely A.C circuit can produce only Joule-heating, and none of the others. So, I made some elementary guess, which, if correct, I think I could use to save time, in competitive-exams, based on multiple-choice questions: _Guess:_   * _Effects, that are directly, or inversely proportional to odd-powers of current through a circuit, are not exhibited in a purely AC circuit, for a complete cycle, and hence, forever._   * _Effects, that are directly, or inversely proportional to even-powers of current through a circuit, are not exhibited in a purely AC circuit, for a complete cycle, and hence, forever._ But I just found out, that in purely capacitive, and inductive AC circuits, no power is dissipated through the respective element, for a complete cycle, and thus, forever,and as well in A.C circuits containing all 3 elements in series- resistor, inductor, and capacitor, power-loss in circuit is zero for a complete cycle, if the current and voltage-drop throgh combination, differ by a phase angle of 90 degrees. The current through these circuits is called **wattless current**. Does the last paragraph oppose my guess?


Could dark matter be expanded atoms, and dark energy expanded radiation?

After watching Lawrence Krauss' lecture 'life, universe, and nothing', an interesting idea lighted up in my mind. Is it possible that dark matter is just spread out matter waves and dark energy extremely long wavelength light? matter -> expansion -> dark matter (delocalized matter waves/huge atoms) visible and xray radiation -> expansion -> invisible light with wavelength greater than the diameter of universe (perhaps the source of energy of the empty space?!!!!) so dark matter today may be what we will become in the future. and we are what dark matter was in the past. eternal expansion just spreads out everything but new stuff is created all the time. anyone living in such a universe at any point of time will see the universe almost as we see it today, expanding with a beginning.


Can somebody explain the graphical integration method in a better way?

After hours of research all what I can find is this: http://www.public.iastate.edu/~fanous/ce332/virtualwork/homepage.html can somebody explain this method in a better way. > If the moment diagram due the applied loads or the moment diagram due to the > virtual load is not continuous, one MUST divide the integration into > segments, each of which is continuous over the integration length.see this > example: The example is somehow hard to understand, Does anybody has a simpler example. I'v been trying to do this method in the conjugate beam method and then trying it with the virtual work method but the results are different. See this example: ![enter image description here](http://i.stack.imgur.com/ot0jD.png) Now assume that $w=50$ and $L=8$. According to the conjugate beam method, the deflection at the center will be -2666.66 but using the virtual concept it is: $\frac{2}{3} \times 400(8)(1.25) = 2666.66$ which is positive, why is this? Also even if I use this method to another load it still give me a different result.


Formation of atoms

If a Proton goes toward an Electron with a trajectory that forms a circular motion, these particles will form an atom ?


Silicon/silicon dioxide interface

Suppose we have an interface of silicon/silicondioxide. If we shine light on the top surface electron-hole pairs are generated in silicon. since there is always a field directing from surface to bulk in silicon (taking n type silicon), electron moves to the bulk and hole moves to the interface states. Is there any change of holes being reached in silicon dioxide surface?


Was the universe a black hole at the beginning?

Big bang cosmology, as far as I understand it, says that the universe was super hot and super dense and super small. It looks like that all the current matter, seen and unseen, were compressed to infinitesimal distance, which means it was a black hole. 1- Is the big bang, and our universe expansion in this case, hence is nothing but a black evaporation via Hawking radiation? 2- Are we living inside that primordial black hole explosion?


Why does gravity need to be quantised?

The electroweak and strong forces seem to be completely different types of forces to gravity. The latter is geometric while the former are not (as far as I'm aware!). So why should they all be described in the same way? - Why are gravitons needed? Alternatively, I suppose my question is why a theory of everything has to be some form of quantum gravity. Is there a reason why a theory of everything requires gravity to be quantised? Is it possible for the quantum world and gravity to be described by some overarching theory which does not require gravity to be quantised? Or is there some reason why this is not the case?


References on the non-compositeness of the known elementary particles

What paper(s) or theory(s) describe or prove that the elementary particles that we have determined today cannot be made up of smaller more fundamental particles?


Has Cosmological Natural Selection been disproved?

I've been reading Lee Smolin's Life of the Cosmos. Great book and it makes a lot of sense that the conditions in black holes are the same as conditions at the big bang. Question is, has his theory about Cosmological Natural Selection been disproved as the wikipedia article states?


What is a good reference for the mechanical properties of materials at cryogenic temperatures?

I want to research various the properties of various materials, namely tensile strength at cryogenic temperatures (~20K). I was wondering if a freely available database of material properties exists.


Newton's color Disk

How does Newton's color disk work? Newton's disk - Take a circular white color disk, make 7 equal intersections and paint section with respective VIBGYOR colors, now when you spin the disk in certain speed, it appears white.


In what sense do Goldstone bosons live in the coset?

Goldstone's theorem says that if a group, $G$, is broken into its subgroup, $H$, then massless particles will appear. The number of massless particles are given by the dimension of the coset, $G/H$. It is then often said that the Goldstone boson's live in the coset. In what sense is this statement true? The Lagrangian is not invariant under transformations of the coset so what does this "living" explicitly mean? To be explicit we can consider the linear sigma model: \begin{equation} {\cal L} = \frac{1}{2} \partial _\mu \phi ^i \partial^\mu \phi ^i - \frac{m ^2 }{2} \phi ^i \phi ^i - \frac{ \lambda }{ 4} ( \phi ^i \phi ^i ) ^2 \end{equation} We define, \begin{align} & \phi _i \equiv \pi _i \quad \forall i \neq N\\\ & \phi _N \equiv \sigma \end{align} and give $\sigma$ a VEV. The spontaneously broken Lagrangian is, \begin{equation} {\cal L} = \frac{1}{2} \partial _\mu \pi _i \partial ^\mu \pi _i + \frac{1}{2} ( \partial _\mu \sigma ) ^2 - \frac{1}{2} ( 2 \mu ^2 ) \sigma ^2 - \lambda v \sigma ^3 - \frac{ \lambda }{ 4} \sigma ^4 - \frac{ \lambda }{ 2} \pi _i \pi _i \sigma ^2 - \lambda v \pi _i \pi _i \sigma - \frac{ \lambda }{ 4} ( \pi _i \pi _i ) ^2 \end{equation} The Goldstone bosons, $\pi_i$, exibit a $O(N-1)$ symmetry, but this is not the coset group symmetry. So where in the Lagrangian do we see this symmetry?


Peak at zero in one device and not the other

I was wondering if anyone could shed some light on this problem. I have placed two accelerometers on an animal one sampling at 50 Hz the other at 100 Hz. They were placed in the same position. I then computed the FFT on my results and found that the spectra from the device sampling at 100 Hz had a large peak at zero whilst the other device didn't display this large zero peak. Any ideas why this might be? I thought that it may be because of the increased sampling rate thus displaying more points that were the same or 'stationary'.


Balloon, lighter than air and vacuum?

As I understand, in a balloon/air balloon, a gas burner is used to heat air or using some lighter atom like helium. Since helium or hot air is lighter and less dense than the cool air around the balloon, the heated air or helium causes the whole balloon to rise. And as I understand, vacuum is a space where there are no atoms inside. So no atom inside means it's lighter than to have atom an inside? Isn't it? So why we can't make something fly by giving it a vacuum container?


Brans-Dicke gravity and Black-Hole "vacuum"

The field equations of the Brans-Dicke gravity are $$\Box\phi = \frac{8\pi}{3+2\omega}T$$ $$G_{ab} = \frac{8\pi}{\phi}T_{ab}+\frac{\omega}{\phi^2} (\partial_a\phi\partial_b\phi-\frac{1}{2}g_{ab}\partial_c\phi\partial^c\phi) +\frac{1}{\phi}(\nabla_a\nabla_b\phi-g_{ab}\Box\phi)$$ In general relativity, the singularity in the Schwarzchild black hole seems to be considered either a 'delta function' like mass source term, or the "edge"/boundary condition of a manifold. Either way, the equations can't tell us how the "infinite curvature"/singularity evolves and therefore we are forced in some sense to "impose it" as a boundary condition either way. However, in Brans-Dicke gravity the effective gravitational coupling constant appears to be sourced by $T$. So it appears in a true vacuum $T \rightarrow 0$, $\phi \rightarrow 0$, and therefore no Schwarzchild solution is possible _regardless_ of the value of the parameter $\omega$. But I've heard that Brans-Dicke is equivalent to General Relativity at least in some limit, so what is going on here? Brans-Dicke seems to really care precisely how we deal with singularity, while General Relativity doesn't care what is behind the event horizon. **1) Can Brans-Dicke gravity support black hole vacuum + curvature singularity solutions?**   **2) Does it force us to interpret the singularity a certain way, and if so does the Brans-Dicke $\longrightarrow$ General Relativity limit tell us the singularity should be interpretted as non-vacuum to make sense?**


Teleportation by quantum isolation

This is a thought experiment, so please don't treat it too harsh :-) Short: If we could isolate two places A and B in the universe from all and any interaction with the surroundings, is there a physical law which states "if something is dropped in place A, it has to stay there"? Long version: Let's assume that the energy of the whole universe is fixed. Let's further assume that it is (by some trick) possible to completely isolate a box of 1m^3, say in the center of a planet (all gravitational and centrifugal forces cancel themselves out), the mass of the planet shields against radiation and we use a trick to shield against neutrinos or we ignore them since the rarely interact with matter). How does an object behave when it has no interaction with the rest of the universe whatsoever? If I put an object in a box described above and I have several such boxes, would it matter in which box the object is? Is there a law which says "even if no one knows, the object **has** to stay where it is"? Or is that just our expectation based on everyday experience?


What is average life in radioactivity and what is its significance?

By definition, average life of radioactive sample is the amount of time required for it to get decayed to 36.8% of its original amount. But what is the significance of 36.8% and why has that value been chosen?


Evaluate Commutator with Partial Derivatives

I need to evaluate the following commutator... $[x(\frac{\partial}{\partial y})-y(\frac{\partial}{\partial x}),y(\frac{\partial}{\partial z})-z(\frac{\partial}{\partial y})]$ i tried applying an arbitrary function $\psi(x)$ but I am unsure of how to evaluate the whole thing. My first step was: $=(x(\frac{\partial}{\partial y})-y(\frac{\partial}{\partial x}))*(y(\frac{\partial}{\partial z})-z(\frac{\partial}{\partial y}))\psi(x)-(y(\frac{\partial}{\partial z})-z(\frac{\partial}{\partial y}))*(x(\frac{\partial}{\partial y})-y(\frac{\partial}{\partial x}))\psi(x)$ This is where i am lost, I am unsure of where to go from here. Maybe there is something else I can do as my first step to simplify the question? please let me know what I can do. Thanks.


How can the nucleus of an atom be in an excited state?

An example of the nucleus of an atom being in an excited state is the Hoyle State, which was a theory devised by the Astronomer Fred Hoyle to help describe the vast quantities of carbon-12 present in the universe, which wouldn't be possible without said Hoyle State. It's easy to visualise and comprehend the excited states of electrons, because they exist on discrete energy levels that orbit the nucleus, so one can easily see how an electron can excite from one energy level into a higher one, hence making it excited. However, I can't see how the nucleus can get into an excited state, because clearly, they don't exist on energy levels that they can transfer between, but instead it's just a 'ball' of protons and neutrons. So how can the nucleus of an atom be excited? What makes it excited?


Why the CFL light bulb flashes even if there is a power cut?

Today when there was a power cut I saw that the CFL light bulbs were flashing time to time at regular interval. I was amazed as there was no source of power to the CFL bulbs even they were flashing as if they were sucking energy from surrounding, collecting it and then releasing them with a flash. Why was it happening?


What would be physics of a star totally consisting of electrons?

What is the minimum radius and mass (if any) for such a structure to be stable? What is the ultimate future of such structure? Will the electron repellation always be stronger than any gravitational attraction? Even at any density?


Can magnets rotate infinitely?

There are many videos on youtube in which people arranged magnets in circle and rotated one placing in middle of that circle on a shaft, and the magnet (magnet motor) starts madly and continues its movement.   Do they really rotate infinitely?   If so, do they from where they get that extra energy to move so fast and infinitely? EDIT: This question is about real world scenario UPDATE: I am unable to find that particular video about which i talked about -- I added them to my favourites, but they are all now deleted :( -- there are hundreds of that that kind of videos. Some example videos links   http://www.youtube.com/watch?v=dT9s33X9D4I   http://www.youtube.com/watch?v=yei0NMqUaZ8   http://www.youtube.com/watch?v=5DvPLFX5gwA   http://www.youtube.com/watch?v=i0oUaPZ_wF8   http://peswiki.com/index.php/OS:Screw-Magnet_Motor   http://www.youtube.com/watch?v=aHVBu77jz4w


Are perpetual magnetic motors a scam?

I read articles and watched videos and more videos where people say they've built perpetual magnetic motors. I do not see any proof that this can work and also I do not see proof that it is wrong. I'm not really into physics. People tend to say this is scum and cannot happen. Can somebody explain what is wrong in the first video?


Lenz's law and bar magnet falling from some height above

I am reviewing my electromagnetism studies and I became confused with where energy of induced emf due to Faraday/Lenz' law come from. Suppose that there is a curved track that starts with some height above loop of wires. The path passes through the center of the loop at the lowest point of the track. The track is symmetrical and it also ends at the same height as the starting height. Now the bar magnet is placed at the starting point. As there is gravitation potential energy converted to kinetic energy, magnetic flux through the loop will change causing an induced emf. But this suggests that by conservation of energy, this electrical energy must have come from somewhere. Otherwise, we would be able to create electrical energy forever using a bar magnet. This is how my confusion arises. Where does the energy come from?


Physics of Snow Globe

What is the physics behind snow flakes inside a 3D snow globe? If I were to implement a snow globe in computer graphics what kind of model do I need to the flakes motion like a real snow globe? What field of physics does this problem belong to?


How to find the force to cause certain trajectory?

So I got a little board today and decided to do some physics. What I came up with is confusing me alot. Say someone throws a rock and when this happens you get the known's of. From here is there a way to find the force that took it to have that trajectory. (kind of like http://imgur.com/2ozNMgH) Angle it was thrown at. (45 degrees) The weight of the rock. (.1 kilo) The distance that the rock was thrown. (82.9056m) Missing Time in Air. Force it was thrown at. Trying to find Speed it was thrown at/force. So with this information we can't really get alot at first but I can almost feel a solution at the top of my tongue. I think that the most critical piece of information in this situation is the angle at which it was thrown (being 45 degrees). From this you can get that the force was split equally between the Y and X component of the situation. I have racked my head for a possible equation (not very advanced in physics of math only high school freshman). All I can think of is that there is some sort of calculus wizardry behind finding this (possibly integral?). I really really have a feeling this can work. Sense I don't know how to write the fancy equations on this and I don't know how to explain what I am thinking enough to explain well. Maybe this will help some more http://www.imgur.com/LHfQJhZ) I understand that because the normal D=1/2gt^2 that there would be two different answers due to it being a quadratic but with it being at 45 degrees they would both be the same. Any sort of help towards finding the answer would be greatly appreciated. Like the old saying,"the fun is getting there, not being there".


Loop-the-Loop with Friction

Let's consider a track that begins vertically becomes a 450 degree loop, and level off. (See diagram) ![enter image description here](http://i.stack.imgur.com/Kuy52.png) We drop a block from height $H$ that falls and goes around the loop. Ignoring air resistance, but taking into account a friction coefficient of $\mu$ against the track, find the smallest $H$ for which the block will complete the loop. By conservation of energy I have obtained the following equation: $$ mgH = \frac{m v(\theta)^2}{2}+mgR(1+\mu-\sin(\theta)) + m\mu R \int_0^\theta\left( g \sin(\theta) + \frac{v(\theta)^2}{R}\right) d\theta $$ Where $v(\theta)$ the the block's speed at $\theta$, and $\theta$ is an angle measured between the block's position (centred at loop center) and a horizontal line going left-wards. The left hand side is gravitational potential energy, the first term on the right is kinetic energy, the next one is also gravitational potential energy, and the last one - the work done by friction. Dividing by $m$, differentiating with respect to $\theta$, and using the chain rule on the first term on the right hand side we obtain: $$ 0=v(\theta)\frac{dv(\theta)}{d\theta} +gR(\mu\sin(\theta) -cos(\theta)) + \mu v(\theta)^2$$ Assuming the block doesn't make it, because it lost's its speed to friction, let's find $\theta$ for which $V(\theta) = 0$. Substituting we get: $$ 0 = \mu \sin(\theta) - \cos(\theta) $$ But it is totally counter intuitive (and incorrect by experiment), that this point depends only on $\mu$ and not on the initial height $H$. So where did I make the mistake? Could somebody please clarify.


How many earth-sized planets have been discovered outside the solar system?

How many earth-sized planets have been discovered outside our solar system? Is there a combined registry of them anywhere? Where might I look for more information?


After the Space Shuttle is retired, how will the new sections of ISS be launched?

The International Space Station (ISS) had many of its components launched by the Space Shuttle. After the Space Shuttle's retirement, what is the proposed launch vehicle for new ISS sections?


The theory of strings stretching between intersecting D-branes

I am trying to understand various aspects of intersecting D-branes in terms of the gauge theories on the worldvolume of the D-branes. One thing I'd like to understand is the worldvolume action for strings stretching between the D-branes. One thing I have considered is $M+N$ D-branes initially coincident but then an angle $\theta$ developing between $M$ and $N$ of them. The gauge symmetry is broken from $U(M\times N)$ to $U(M) \times U(N)$ with the off- block-diagonal terms of the gauge field becoming massive. I anticipate that $\theta$ is the vev of some Higgs field that mediates this transition. What is the action of this Higgs field and where in the string spectrum does it come from?


Why does an octave on a piano have the divisions of 8 white keys and 5 black keys?

An octave corresponds to a doubling or halving of the frequency. Each octave on a piano and in classical music score is broken downs into 8 white keys and 5 black keys. Is there a physics explanation for that division?


The shape of the graph of the equation: $V= -r I + E$

I have recently collected data (for a school experiment) in order to measure the EMF and the internal resistance of a solar cell. The data complied with the equation: $V = -rI + E$, i.e. the voltage decreased as the current increased and I have a negative gradient. However, I am having some trouble as to explaining why the graph is the way it is, that is, why is there a negative gradient? Here's my rationale: Using Ohm's law, we can see that decreasing the (external) resistance increases the current flowing throughout the whole circuit (it is a simple series circuit with just a solar cell, ammeter, variable resistor and a voltmeter (in parallel, obviously)). Now, the voltage drop across the solar cell should increase as well (assuming its internal resistance in constant) because $v = V_r = Ir$, where $v$ is the 'lost volts'. For this reason the voltage drop across the load must decrease(which is what we see from the graph) because $E = V + v$. Thus, the voltage drop across the resistor decreases as current increases. I don't know to what extent this is correct or if it is correct at all but I would appreciate any and all help given. Also, as you can probably tell, the level of physics used here is very basic, so I would also appreciate it if the answers explained it in this way as well, although, where possible, do not sacrifice accuracy for the sake of simplicity.


Is there any case where one would use, snap, crackle or pop?

As we all know, if you differentiate distance with reference to time, you get speed, and likewise, differentiating speed you get acceleration. However, if you keep differentiating, to the rate of change of acceleration and so forth (more emphasis on the so forth): is there any point to this? Does it have any useful application in any industrial or practical industry, as opposed to theoretical physics? Also, what could we possibly learn about a situation by understanding the rate of the rate of the rate of change of the objects distance?


transforming experimental rate constant to monte-carlo rate constants?

I am trying to write a kinetic monte carlo code for polymerization process, but i am confused about how to compare my monte carlo equilibrium with analytical calculations. as an e.g. $X + X <=> X_2$ with a forward rate $k_a^+$ and backward rate $k_a^-$. $X_2 + X <=> X_3$ with a forward rate $k_e^+$ and backward rate $k_e^-$. $X_3 + X <=> X_4$ with a forward rate $k_e^+$ and backward rate $k_e^-$. ............ etc. Now i want to specify concentration of monomer X. volume of the container V and avogadro's number. But what i am confused in is that how to transform experimental rate constants to monte carlo rate constants. real or experimental units for $k_a^+$ are $mol^{-1}s^{-1}$ and $k_a^-$ are $s^{-1}$ and same for $k_e^+, k_e^-$ the equilibrium constant $K_e = \frac{k_e^+}{k_e^-}$ and $K_a = \frac{k_a^+}{k_a^-}$ is usually given in (/mol) units in literature. Nitin


Total Electrical potential energy of two particle system

I recently have been studying Electro-statics and I couldn't understand properly how the potential energy of two particle system is found. > Suppose you have two particles with charges $Q_1$ and $Q_2$ respectively. > The distance between them is $r$. What is the total electrical potential > energy of the system comprising of the two particles? Well, I know the > answer. I want to know the reasons behind the answer. My book says "Fix one of the charges and then bring the other from infinity. Hence we get the answer. I am not satisfied with the answer, as I saw in Wikipedia they find the potential of each other with respect to the other and then add and divide by $2$. Why is this taking potential with respect to each other valid? Why do they divide by two? What does taking potential with respect to other mean? Do they fix one charge and bring another one? what about three charges $Q_1, Q_2$ and $Q_3$ which are situated in the vertices of an equilateral triangle with side length $r$?


Alpha scattering, observations and conclusions

For Rutherford's alpha scattering experiment where he fired alpha particles at a thin sheet of metal to determine the size of the nucleus, please can someone give me the direct link between his observations and his conclusions e.g. this observation lead to this conclusions.


Calculating displacement amplitude of ultrasonic power transducer

I am not a physicist, but my current project drives me to some physics-related computations, hence seeking help. I have some ultrasonic transducers, 5938D-25LBPZT, for which very limited information is available. ![Transducer Diagram](http://i.stack.imgur.com/PCWcM.jpg) ![Transducer Photo](http://i.stack.imgur.com/j5dKc.jpg) In order to define the parameters for an ultrasonic booster and amplification horn for this device, I need to compute the displacement amplitude of its radiating surface. The procedure I am following, and the **base data** ( _or where data is unavailable, my assumptions_ ) are thus:   * Operating power: **50 Watts** RMS (the transducer is rated for 60 Watts)   * Emitting horn material: Stainless Steel **SS316** (best-guess, unfortunately, as SS316 is typical in other such transducers)   * Emitting face diameter: **5.865 cm** (measured)   * Frequency at resonance: **_f_** = **24989 Hz** (measured)   * Best-case power transmission: **90%** (in forward direction, according to various sources) **Constants** :   * Density of SS316: **ρ** = **7.8 gm/cm^3**   * Velocity of longitudinal sound propagation: **C** = **5.8 mm/μSec** (from this reference, approx)   **_Should the axial velocity of sound be considered instead for this calculation?_** **Calculations so far** :   * Face area: **27.016 sq cm** (ignoring area lost to bolt-hole)   * Sound intensity at face: **I** = 90% x Power / Area = **1.666 Watts / sq cm**   * Displacement Amplitude: **A**   ![Amplitude Formula](http://i.stack.imgur.com/dfeib.jpg)   * ... where **ω** = 2 x π x _f_   * Thus, **A = 172.8296514 nanometers** * * * **My questions are:**   1. I am not sure I have managed to convert everything to consistent SI units, so I need help confirming that I've got that part right   2. If anyone has experience in such power ultrasonic piezo transducers, a confirmation that my results are more or less within the realms of plausibility would really help.   3. Is there some significantly incorrect assumption or constant value in the foregoing?   4. Should I be using some online resource that trivially calculates all this from the rated power and other inputs, instead of struggling with it manually? * * * **Update** : _Adding diagrams to show the excitation direction and materials involved._ ![Ultrasonic Stack](http://i.stack.imgur.com/wknTr.gif)   (please ignore _Diamond insert_ ) This is the final arrangement of the Ultrasonic Stack. The _booster_ and _horn_ in the diagram above will both be SS316. In order to compute dimensions for these two parts for resonance, I need the displacement amplitude of the front face of the steel "radiation head" in the transducer ( _first diagram here_ ), which is to be bolted to the booster. The bolt-hole in the radiation head can be seen in the photograph above. Another diagram, for a similar "Ultrasonic Stack", with the Langevin transducer itself split up. This diagram has a cylindrical radiation head, as opposed to the cone section shape in the transducers I need to compute for. ![Ultrasonic Stack](http://i.stack.imgur.com/5zuUG.jpg)   (please ignore _Wedge_ ) (Last two diagrams are from articles on ScienceDirect.com)


How do we see different colours?

* Why do different wave lengths cause electrons to behave(?) differently, causing us to see different colors?   * What is happening at the quantum level which causes the colour black to absorb all of the wavelengths of white light?   * Why is it that a black can of soft drink will heat up more than a white one?    * How does the energy, while passing through different colours, increase the temperature differently?


Wavefunction in first Brillouin zone

We know that with symmetry, Brillouin zone is nothing but copies of its irreducible zone, so can we conclude that we can find all possible wavefunctions in its irreducible zone? What about wavefunctions in high- symmetric points?


What does an atom radiate: a wave packet or a single photon?


Why two connected 1.5 volt battery become 3 volt? If one battery plus side is connect to minus side of another battery, there should be a current flow between them until no free charge move anymore, then the final state of these two side is 0 volt. So the total two connected battery should be still 1.5 volt. This is wrong actually, but I can't think out why.


Motion in insulating fluid under high voltage

I observed the following phenomenon in an experiment (I'm not a student of physics, just an amateur) and was hoping for an explanation. A metal pan is electrically grounded and a layer of insulating fluid (vegetable oil) is placed in the pan. If you like, assume the pan is infinite in diameter. An electrode (a needle) is placed above the pan, pointing downward at the oil. A high voltage (-12kV) is applied to the electrode. The electrode is close enough to the pan to generate an ion wind, but not close enough to overcome the breakdown voltage of the air and oil, so there is no arc. The oil is repelled from the electrode (perhaps because it acquires a negative charge from the ion wind) and also, more remarkably, the surface of the oil begins to undulate. My theory to explain the undulation is that the negatively charged oil on the surface is repelled from the electrode until it touches the pan, where it loses its charge and therefore is permitted to rise to the surface again, and the process repeats. So the undulations are something like convection cells, only with electric charge instead of heat. Is this explanation correct? Is there a name for this phenomenon? Would a similar phenomenon create leakage current in capacitors with liquid dielectrics?


How does one produce entanglement?

In practice, when entangling two particles (say two electrons), is creating an entangled state purely a matter of bringing them close enough together? What I mean is, to make their wave functions correlated, is it just a matter of bringing them close enough together (regardless of what is required to accomplish this), or are there other processes necessary to cause the entanglement?


What are the practical applications of quantum foundations?

Many quantum foundation researchers keep emphasizing that For All Practical Purposes (FAPP), quantum foundations are irrelevant. They even invented an acronym for it! Does that mean that quantum foundations have no practical applications? If so, why bother working on quantum foundations?


How is relativity equation approximated

What is the technique with which I can approximate the equation $$\frac{mc^2}{\sqrt{1-(\frac{v}{c})^2}}-mc^2$$ when $v\ll c$? Any hint would be much appreciated


Is "microbunching" in a free electron laser limited by the Pauli exclusion principle?

As I understand it, a free electron laser can basically be pictured as a synchrotron light source with an undulator which by the particular setup causes the electrons to self-attune so that they produce light which is both coherent and monochromatic (as opposed to the usual synchrotron which produces monochromatic but incoherent light). The process of self-attuning can be also replaced by attuning the electrons by an external laser. Skipping the details, the resonance caused by the right set of parameters causes something which is called "microbunching" in the electron beam, i.e., an effect which can be classically pictured as all the electrons clumping around points in the beam with a distance of one optical wavelength in between. Same spatial points means same undulation, so they oscillate throughout the beam with the same phase causing coherence of emitted light. (I have just read about it, this might be a slightly naive description.) All the electrons at one place and with one velocity? Pauli exclusion principle comes quickly to mind. It seems to me that at least for small wavelengths (say ångströms $\to$ X-rays) the bunching has to be done at the cost of a noticeable velocity dispersion of the electrons due to the Pauli exclusion principle. This would mean a dispersion of the produced wavelengths and possibly a tendency to further dilute the bunched electron packets. Also, a larger required brightness of the output means more electrons and once again a wider peak in the phase space to fit the electrons into. This would in a very hand-waivish manner suggest that there is a certain constraining relation between the wavelength/monochromaticity/coherence/brightness of the outcome. So the question is: **_Does such a fundamental constraint of a free electron laser exist? And is it relevant for the current state of art or the near future?_**


What is the lowest possible theoretical temperature that nuclear fusion can occur at?

I am not talking about the pseudo-science of so called cold fusion I am interested in what temperature you can get away with to produce fusion reaction. I was thinking in terms of micro-fusion or at least at levels below the huge tokamaks being produced because obviously small scale devices couldn't handle millions of degrees of temperature. Or correct me if I am wrong please as I am no expert just a curious student!!


Physical question on an RLC circuit

I do not know if this is an appropriate place to ask this question here but this is the only website that I am a member of so I hope that it is okay. If not, I can delete it. This circuit question was brought to me while tutoring yesterday that I was not able to answer physically. Initially the capacitor had been charged to 2V so $v_C(0^−) = 2V$. Before the switch is thrown there is no current through either the resistor or inductor, so $i_L(0^−) = 0A$. ![enter image description here](http://i.stack.imgur.com/jlIuR.jpg) Since the inductor and capacitor cannot have an instantaneous change in their values just before and after the switch is thrown, these values must remain the same: $i_L(0^−) = i_L(0^+−) = 0A$, $v_C(0^−) = v_C(0^+) = 2V$ This implies that at the moment the switch is closed, the inductor shorts out the capacitor and resistor, and no current occurs along any branches of the circuit. That is, the voltage drop across all three elements is zero. What is happening to the capacitor’s and inductor’s current and voltage in the circuit? How can the capacitor have a voltage drop of 2V and yet the inductor appears to short it out simultaneously?


When polarized light hits a polarizer, what happens to the polarization which is not transmitted?

It is typically said that in polarization of light only those waves can pass through a polaroid which has a plane of vibration parallel to the axis of polaroid. My question is: what happens to the other waves (the other polarization)?


Why does the windshield of my car freeze even if the outside temperature is above freezing?

Under what conditions does the windshield of a car freeze even if the outside temperature is above freezing? It is not clear whether this is related to the question why bridges freeze with non-freezing outside temperatures (see related topic), as I did not see an explanation in terms of radiation there. It indeed seems that electromagnetic radiation is the culprit for windshields, which would predict that this phenomenon can only occur with an open sky. Does anybody know whether this is true? And why does ice only form on top of the windshield and not on the metal surface of the car? Is this because heat is conducted much faster on the metal surface?


Why does frequency increase as the length of an open air column shortens?

I am curious as to why the frequency of a wavelength increases as an open air column becomes smaller in length. Is it because an open air column will always contain half a wavelength, therefore if the tube is smaller, it means a shorter wavelength, which in itself means a higher frequency?


Why does a flat clover-like shape fall slower when it is rotating?

The plastic flat clover-like shape pictured below falls noticeably slower when it is rotating fast. I wonder why. Its three edges are flat, so I don't think that the rotation makes it act like a propeller. An ideal explanation would include an analogy that a young child can understand (e.g. rain bouncing on a car's wind-shield). Here is a photo of the object. ![object that falls slower when rotating](http://i.stack.imgur.com/i9pNt.jpg)


Why do we see laser beams?

A laser produces a coherent beam of photons with particular $\mathbf{k}$ and $\mathbf{\omega}$. So, if there is no particular $\mathbf{k}_0$ directed toward our eyes, why do we see laser beams? I think this is related to the Huygens–Fresnel principle, but I wasn't sure.


Magnetic moment precession around magnetic field

I have a question regarding the magnetic moment of an atom. If you just have one atom, and you apply an external magnetic field in lets say the z-direction. Then the magnetic moment will precess around the direction of the magnetic field applied, and depending on the quantum numbers it can precess in different quantized directions. The question is rather simple: WHY does it precess around the magnetic field ? My book only states that it do indeed precess if a magnetic field is applied, but as I'm asking, I'm not sure WHY it does ? Why can't it just point in one direction all the time ?


Do we need infinite energy to make 2 similar charges touch only in theory?

By Coulomb's law, say if we have 2 point particles each having a charge of +1C then by the formula, F = k/(d)^2 if we need to make the distance between them zero, clearly y the formula, we need to have an infinite amount of force. However, in a real life scenario, we see that if we rub two balloons with our head, they acquire the same charge. However, it is still possible to make them touch. Is this because the two balloons are not point objects? OR is the definition d=0 actually mean "to occupy the same space". for d to be 0, do the objects only need to touch or do their centers need to be together?


Examples of circuitry using proton currents

Proton cuircuits and proton motive force are part of standard discussion in biology and processes involving photosynthesis. The sort of proton currents discussed in biology are obviously slightly different than the magnetically confined proton currents in an acclerator like LHC, however, although we are very familar with electronic circuitry, I was wondering if there are example of protonic circuitry outside of biological processess.


Is it Possible to have Adiabatic Processes other than $PV^\gamma$ for the ideal Gas?

Is it possible to represent an adiabatic process for an ideal gas by a formula other than $PV^\gamma=Const$?: _Relevant Considerations_ : We always need to connect a pair of _arbitrary_ points/states $A:(P_1,V_1)$ and$B:(P_2,V_2)$ on the P-V indicator diagram to define the Internal Energy function(U) If the two points do not lie on some $PV^\gamma=Const$, we may think of some abrupt process connecting the two states. Such a process cannot be represented by a continuous curve on the diagram since the intermediate states are non- equilibrium states. Pressure will not have a unique value wrt the system for such processes.It will be different from point to point.We have a pair of points A and B without any curve connecting them Now we connect the points A and B by an arbitrary "continuous" path on our diagram(on paper) and select close discrete points on it. The consecutive "close" discrete points may be connected by abrupt adiabatic "processes" as described previously. Therefore in the pragmatic sense we can have an arbitrary quasi-static "adiabatic" path connecting the points A and B : The "close" points ie physical; states,will lie on a curve different from $PV^\gamma=Const$ [In the physical/practical implementation of the $PV^\gamma=Const$ curve we do consider/implement a chain of discrete equilibrium states]


Polchinsky's Evaluation of the One Loop String Path integral

I try to evaluet the matrix M in the Polchinsky's article(Communications in Mathematical Physics,1986, Volume 104, Issue 1, pp 37-47,"Evaluation of the one loop string path integral",Joseph Polchinski) (1)$$\det\begin{bmatrix}1 & 0 & 0\\\ -D_{a} & \delta_{a}^{c} & 0\\\ \frac{1}{2} g^{ef}g_{ef,i} &0 & \delta_{ik}\end{bmatrix} \cdot \det\begin{bmatrix}2+4C & 0 & 0\\\ 0 & 2 \Delta_{c}^{d} & -2D_{e} {\chi_{l}^{e}}_{c}\\\ 0 &2 \chi_{k}^{ed} D_e & \chi_{kef} \chi_{l}^{ef}\end{bmatrix} \cdot\det\begin{bmatrix}1 & D^{b} & \frac{1}{2} g^{ef}g_{ef,i}\\\ 0 & \delta_{d}^{b} & 0\\\ 0 & 0 & \delta_{ij}\end{bmatrix}=J N J^{T}$$ where $\Delta_{c}^{d}= -\delta_{c}^{d} D^2-D^{d}D_{c}+D_{c}D^{d} and \chi_{iab}=g_{ab,i}-\frac{1}{2} g_{ab}g^{cd}g_{cd,i}$. I'm considering a torus, so the covariant derivatives become $\partial$. Also the determinant of $J$ and $J^T$ are $1$. So the determinant of the matrix N become $$\det(N)=\det(2+4C)\det(-2\delta_{c}^{d}\partial^2) \det(\chi_{kef} \chi_{l}^{ef}).$$ This is right? (2) $\det(\chi_{kef} \chi_{l}^{ef})=\chi_{kef}(g^{e \alpha}g^{f \beta} \chi_{l\alpha \beta})=\partial_k g^{\alpha\beta}\partial_l g_{\alpha \beta}-\frac{1}{2} (g_{\alpha \beta}D_kg^{\alpha \beta})g^{\gamma \delta} \partial_l g_{\gamma \delta}-\frac{1}{2} g^{\alpha \beta} \partial_l g_{\alpha \beta} g^{cd} \partial_k g_{cd}+\frac{1}{2}(g^{cd}\partial_k g^{cd})(g^{\gamma \delta} \partial_k g_{\gamma \delta})$ this is also right? (3) If steps 1 and 2 are correct, how can I get to the form (given by Polchinski in the article)? $$\det'(N)=\det(2+4C)\det'(-2\delta_{c}^{d} g^{ab} \partial_a \partial_b) \frac{4}{{\tau_2}^4}$$ where the prime denotes the restriction to the zero mode.


Is the Copenhagen interpretation still valid?

"The secret lives of photons revealed" - physicsworld.com. It seems that this experiment violates the Copenhagen interpretation. Is it still valid? Can you please give me more insight on this experiment?


Is it possible to have a negative attraction to gravity just like how magnets with same pole does?

Layman question here, is it possible for an object to be negatively attracted to a gravitational pull of another object? Like how same pole magnets repels each other, can it happen on gravity too?


Is it possible to Vectorialize Quantum Field Theories?

If I take the rules for classical electrodynamics in the covariant formulation (the closest to QFT), I have a tensor that describes the field, $F_{\mu\nu}$. Now we know that we can take some of the components of that tensor and find two vectors, $\mathbf{E}$ and $\mathbf{B}$, that obey Maxwell equations and have an identity on their own. In my understanding the "photon field" has been reduced to something much more material, that I can visualize. I can see easily, for example, a wave propagating in a particular direction. I am aware that I tacitly assumed that the "photon field" is described by the classical theory, but I think that the concept is clear nonetheless. Now, is it possible to do something like that to the "electron field", the "positron field", the "gluon field" and so on? If you take all the forces together and you have a giant field with many components (as in my understanding is the Standard Model), is it possible to find some combinations of those components that behave like vectors?


Hydrogen Transition Rate from n',l to n',l'

I am trying to solve Problem 9.22 in Griffiths' Intro to Quantum Mechanics (2nd Ed.) that asks to show the spontaneous emission rate for a transition from n,l to n',l' in hydrogen is given by $\frac{e^2\omega^3I^2}{3\pi\epsilon_0\hbar c^3} \begin{cases}\frac{l+1}{2l+1} & if l'=l+1 \\\ \frac{l}{2l-1} & if l'=l-1 \end{cases}$ where $I=\int_0^\infty r^3R_{nl}(r)R_{n'l'}(r)dr$ I found this question rather interesting because it generalizes transitions for the hydrogen atom and that the result is independent of m. Why should this be the case? Attempt at solution. As shown below, I am having difficulty adding everything up to prove the desired result above. The spontaneous emission rate for a transition from state a to state b is given by $\frac{e^2\omega^3|p|^2}{3\pi\epsilon_0\hbar c^3}$ where $p=q\langle b|r|a\rangle $ (the text notes that $p$ in general may be complex but is treated as real). The hint in the text suggests that you integrate over the x,y and z coordinates between $|nlm\rangle$ and $|n'l'm'\rangle$, and then calculate $|\langle n',l+1,m|\boldsymbol{r}|n,l,m\rangle|^2+|\langle n',l+1,m-1|\boldsymbol{r}|n,l,m\rangle|^2+|\langle n',l+1,m+1|\boldsymbol{r}|n,l,m\rangle|^2$ This hint is based on section 24 of Fermi's Notes on Quantum Mechanics (The spherical harmonic relations shown below including conversion to Cartesian coordinates are obtained from this section. The solution to the integrals shown below for the case $l'=l+1$ perhaps unsurprisingly agree with those in this text. The integrals are not provided for the $l'=l-1$ case). In order to make this hint meaningful, we first need to remember (or in my case look up) some relations for spherical harmonics: $\sqrt{\frac{8\pi}{3}}Y_{1}^{1}Y_{l}^{m-1}=\sqrt{\frac{(l+m)(l+1+m)}{(2l+1)(2l+3)}}Y_{l+1}^{m}-\sqrt{\frac{(l-m)(l+1-m)}{(2l+1)(2l-1)}}Y_{l-1}^{m}$ $\sqrt{\frac{4\pi}{3}}Y_{1}^{0}Y_{l}^{m}=\sqrt{\frac{(l+1)^{2}-m^{2}}{(2l+1)(2l+3)}}Y_{l+1}^{m}-\sqrt{\frac{l^{2}-m^{2}}{(2l+1)(2l-1)}}Y_{l-1}^{m}$ $\sqrt{\frac{8\pi}{3}}Y_{1}^{-1}Y_{l}^{m+1}=\sqrt{\frac{(l-m)(l+1-m)}{(2l+1)(2l+3)}}Y_{l+1}^{m}-\sqrt{\frac{(l+m)(l+m+1)}{(2l+1)(2l-1)}}Y_{l-1}^{m}$ We will also need some rather useful relations between spherical harmonics and Cartesian coordinates: $\sqrt{\frac{8\pi}{3}}Y_{1}^{1}(\theta,\phi)=-\frac{x+iy}{\sqrt{x^{2}+y^{2}+z^{2}}}$ $\sqrt{\frac{8\pi}{3}}Y_{1}^{-1}(\theta,\phi)=-\frac{x-iy}{\sqrt{x^{2}+y^{2}+z^{2}}}$ $\sqrt{\frac{4\pi}{3}}Y_{1}^{0}(\theta,\phi)=-\frac{z}{\sqrt{x^{2}+y^{2}+z^{2}}}$ We can see from looking at the above equations that the integrals with z will be nonzero when the m values are equal while the integrals containing $x\pm iy$ will be nonzero when the m values are different (i.ei $m'\neq m$). First let us focus on the $l'=l+1$ case. From the above relations we can see that $\langle n,l+1,m+1|x+iy|n,l,m\rangle=-I\sqrt{\frac{(l+m+2)(l+1+m)}{(2l+1)(2l+3)}}$ $\langle n,l+1,m+1|x-iy|n,l,m\rangle=0$ $\langle n,l+1,m|z|n,l,m\rangle=I\sqrt{\frac{(l+1)^{2}-m^{2}}{(2l+1)(2l+3)}}$ $\langle n,l+1,m-1|x+iy|n,l,m\rangle=0$ $\langle n,l+1,m-1|x-iy|n,l,m\rangle=I\sqrt{\frac{(l+1-m)(l+2-m)}{(2l+1)(2l+3)}}$ Now let us add up the above elements. This is where I am stuck. I obtain $\langle n',l',m'|r|n,l,m\rangle=|\langle n,l+1,m+1|x+iy|n,l,m\rangle|^{2}+|\langle n,l+1,m|z|n,l,m\rangle|^{2}+|\langle n,l+1,m-1|x-iy|n,l,m\rangle|^{2}$ $\langle n',l',m'|r|n,l,m\rangle=I^{2}\left\\{ \frac{(l+m+2)(l+1+m)}{(2l-1)(2l+3)}+\frac{(l+1)^{2}-m^{2}}{(2l-1)(2l+3)}+\frac{(l+1-m)(l+2-m)}{(2l+1)(2l+3)}\right\\} $ $\langle n',l',m'|r|n,l,m\rangle=I^{2}\left\\{ \frac{l^{2}+lm+2l+l+m+2+lm+m^{2}+2m}{(2l-+1)(2l+3)}+\frac{(l^{2}+2l+1)-m^{2}}{(2l-+1)(2l+3)}+\frac{l^{2}+l-lm+2l+2-2m-lm-m+m^{2}}{(2l+1)(2l+3)}\right\\} =I^{2}\left\\{\frac{(3l^{2}+8l+5+m^{2})}{(2l+1)(2l+3)}\right\\} $ which is clearly not equal to what we want. I apologize if I simply do not know how to add, but the extra $m^2$ makes me believe there is some trick that I am missing. In any case, we may continue on for the $l'=l-1$ case. Similar to the above, we obtain $\langle n,l+1,m+1|x+iy|n,l,m\rangle=I\sqrt{\frac{(l-m-1)(l-m)}{(2l+1)(2l-1)}}$ $\langle n,l+1,m+1|x-iy|n,l,m\rangle=0$ $\langle n,l+1,m|z|n,l,m\rangle=-I\sqrt{\frac{l^{2}-m^{2}}{(2l+1)(2l-1)}}$ $\langle n,l+1,m-1|x+iy|n,l,m\rangle=0$ $\langle n,l+1,m-1|x-iy|n,l,m\rangle=-I\sqrt{\frac{(l-1+m)(l-2+m)}{(2l+1)(2l-1)}}$ Now let us add the above together like we did for the previous case. With any luck, we will get stuck on this part again. $I^2\left\\{ \frac{(l-m-1)(l-m)}{(2l+1)(2l-1)}+\frac{l^{2}-m^{2}}{(2l+1)(2l-1)}+\frac{(l-1+m)(l-2+m)}{(2l+1)(2l-1)} \right\\}$ $=I^2\left\\{ \frac{l^2-lm-l- lm+m^2+m}{(2l+1)(2l-1)}+\frac{l^{2}-m^{2}}{(2l+1)(2l-1)}+\frac{l^2-l+lm-2l+2-2m+lm-m+m^2}{(2l+1)(2l-1)} \right\\}$ $=I^2\left\\{ \frac{3l^2-4l+2-2m+m^2}{(2l+1)(2l-1)}\right\\}$ Again, we appear to get stuck. Thank you for reading this post to the end. Any help would be greatly appreciated in eliminating the m variable from the above sum and proving the answer.


Need help solving a task - laws of conservation (leptons, baryons, charge)

We got a task from our teacher, a multiple choice task; and here it is: Which of the following reactions is correct, considering the conservation laws? Note: **We have only learned to use the conservation laws for Electric charge, Baryon number and Leptonic number.** ![The Tasks:](http://i.stack.imgur.com/BQRl2.png) However, i found out that **B** and **C** must be wrong regarding to those laws, and that **A** and **D** is both correct. But the task ask specificly about the correct answer, and i am stuck here. And please also, explain why one of them should be wrong, regarding to those 3 laws, only. My teacher had problems solving it aswell, so it might be an error in the task itself. Thankyou!


Are the electrons in a quantum hall edge state entangled?

I am reading the paper on Quantum Energy Teleportation by Yusa, Izumida and Hotta(This article), and it seems that they are assuming that the quantum hall edge state is a quantum correlated state, which I understand as it being an entangled state. How can all the electrons in the edge state be entangled? Or have I misunderstood the assumptions of the article?


Angular momentum conservation in a central field through the Hamiltonian

In my teacher's notes there is a discussion of the Hamiltonian for a central force field with potential $V(r)$. The Hamiltonian is formulated in spherical polar coordinates: $$H=\frac{p_r^2}{2m}+\frac{p_\phi^2}{2mr^2\sin^2\theta}+\frac{p_\theta^2}{2mr^2}+V(r)$$ Then the conservation of $L_z$ is trivial, because $\phi$ is cyclic. However it is asserted, as if it were evident, that $L^2$ is also conserved. One consequence is that the Hamiltonian can be written in the form $H=\frac{p_r^2}{2m}+\frac{L^2}{2mr^2}+V(r)$. Now, I know that for any radial force it is trivial to prove that $\vec L$ is conserved, but how can you prove that $L^2$ is conserved just from looking at the previous Hamiltonian? Moreover, does it suffices to prove that $L^2$ and $L_z$ are conserved in order to find that all the three components of $\vec L$ are conserved? I'm a bit confused, so any help is appreciated