Huygens Principle and principal of rectilinear propagation of light

Suppose I have an wave source and light waves are radiating from it. If I have a point source, then after a time t, with a radius of ct I will have a circular wave front.By Huygens principle each point on the wave front acts like a secondary source. From this point again light can travel in all directions. So isn't the principle of rectilinear propagation of light violated?![enter image description here](http://i.stack.imgur.com/ipOCi.png) Consider the light following the specified path shown with the arrow head.


Is the density operator a mathematical convenience or a 'fundamental' aspect of quantum mechanics?

In quantum mechanics, one makes the distinction between mixed states and pure states. A classic example of a mixed state is a beam of photons in which 50% have spin in the positive $z$-direction and 50% have spin in the positive $x$-direction. Note that this is not the same as a beam of photons, 100% of which are in the state $$ \frac{1}{\sqrt{2}}[\lvert z,+\rangle + \lvert x,+\rangle]. $$ It seems, however, that at least in principle, we could describe this beam of particles as a single pure state in a 'very large' Hilbert space, namely the Hilbert space that is the tensor product of all the $\sim 10^{23}$ particles (I think that's the proper order of magnitude at least). So then, is the density operator a mathematical convenience, or are there other aspects of quantum mechanics that truly require the density operator to be a 'fundamental' object of the theory? (If what I mean by this is at all unclear, please let me know in the comments, and I will do my best to clarify.)


How much energy is necessary to set a year to exactly 360 days

How much energy would be necessary to slow down Earth rotation such that a year was 360 days long? In the same spirit: how much energy would be necessary to make the Earth rotate faster around the Sun such that a year was 360 days long? _(Just for fun: how would you do it practically?)_


How to calculate the electric field at a point in space

Let's say I have a uniformly-charged wire bent into a semi-circle around the origin. How can I find the electric field (magnitude and direction) I'm not even sure if I should use Coulomb's or Gauss' law either. Kind of stuck setting up a solution and starting somewhere.


I got asked this question for my physics course at college, i'm not the best at airspeed velocities... > What is the airspeed velocity of an unladen swallow? I still can't find anything to help me, i'm quite stuck and have been googling and asking about on a few forums. Best help I found was this page on the subject. Thank you.


Does the Bohr van Leeuwen Theorem also apply to ferromagnetism?

I know that the Bohr-van Leeuwen theorem shows that there could be not consistent pure classical explanation of dia- and paramagnetism. Does the same theorem also rule out a consistent classical theory of ferromagnetism? Do you have any realiable references for this?


Does closing curtains 'make your home warmer'?

I mean, in the sense that the act of closing curtains would somehow reduce the amount of heat loss of the house to the outside, thus making it warmer _for a given supply of heating_.


Symmetry breaking under isothermal expansion

Is there any example of a symmetry breaking phase transition in a system of particles under isothermal expansion?


How to reconstruct information from a graph of an oscillation?

We are given a graph of the position of a wave (amplitude). How can we calculate the wavelength, frequency and the maximum speed of a particle attached to that wave? We have _Speed = wave length $\times$ frequency_ , _$W=2 \pi \times$ frequency_ , $V_{max}=A\times W$. So how to calculate A? ![enter image description here](http://i.stack.imgur.com/9vkUA.png)


Electrons and energy

I'm an a-level student. While studying quantum physics I came across this statement:"According to electromagnetic theory an accelerating charge emits energy". The bottom-line was that an electron (which is an accelerating charge in an atom) should, according to this, emit electromagnetic radiation, but it does not. Can someone please explain the electromagmetic theory and why the electron does not emit energy? Please keep in mind I'm only an a-level student.


Meaning of terms and interpretation in the electric multipole expansion

In section 3.4.1 of Griffiths' _Introduction to Electrodynamics_ , he discusses electric multipole expansion. He derives the formula or the electric potential of a dipole, which I follow, but right after, he begins talking about the electric potential at a large distance, which is as follows: $$V( \vec{r}) = \frac{1}{4 \pi \epsilon _{0}} \int \frac{\rho (\vec{r}')}{ℛ} dV$$ $ℛ$ is from some point inside the charge distribution to the point P. **What exactly does $\vec{r}'$ denote?** Is it the distance from the center of the charge distribution? Next, he uses law of cosines to find the expression for $ℛ^{2} = r^{2}+(r')^{2}-2rr'\cos\theta'$. This gives the same question as above, what does the symbol $r'$ mean? Then he defines $ℛ = r(1+ \epsilon)^{1/2}$, where $\epsilon \equiv \left(\frac{r'}{r}\right)^{2}\left(\frac{r'}{r}-2\cos\theta^\prime\right)$. The proceeding part is where I really get lost: > For points well outside the charge distribution, $\epsilon$ is much less > than 1, and this invites a binomial expansion. $$\frac{1}{ℛ} = > \frac{1}{r}\left[1- (1/2) \epsilon+ (3/8) \epsilon ^{2} - (5/16) \epsilon > ^{3}+\ldots\right]$$ **What is going on in this last step?** And finally, we eventually derive the formula $$V(\vec{r}) = \frac{1}{4 \pi \epsilon _{0}} \sum ^{\infty}_{n=0}\frac{1}{r^{n+1}} \int(r')^n\,P_{n}(\cos \theta)\,\rho( \vec{r}')dV $$ Why did we go for all this trouble? If this is supposed to be the electric potential at a large distance, **couldn't we have just used $V( \vec{r}) = \frac{1}{4 \pi \epsilon _{0}} \int \frac{\rho (\vec{r}')}{ℛ} dV$**? I don't understand what this equation _actually means_ in physical terms. And in regards to the actual equation, **what is $P_{n}$**? Any help is much appreciated.


How is energy extracted from fusion?

I understand that combining deuterium and tritium will form helium and a neutron. There are three methods to do this (1) tokamak (2) lasers and (3) cold fusion. I would like to know after helium is formed. How is that energy extracted from tokamak and stored?


What is wrong in this representation of relative reference systems?

Up until now I've explained relative time to myself as looking at the 3D world from different four-dimensional perspectives, analogous to how looking at a 2D-ish object (eg. a sheet of paper) from different angles makes it appear to change shape. Also, I've used this representation below to explain how the time at which an event occurs could change depending on the body's reference system. This representation uses two bodies that move relative to each other, $B1$ and $B2$, to represent the time of occurrence of events $E1$ and $E2$. ![](http://i.stack.imgur.com/KsZ2r.png) ( _Not very precise -- blame MS Paint :P -- but my intent should be clear._) I am not sure, but as far as I know $t(E1) \le t(E2)$ should hold true for any reference system (order should be preserved), which means that the drawing is correct. But I just realized that in certain cases, it is possible for $E2$ to occur before $E1$ (according to my representation): ![](http://i.stack.imgur.com/8X5n9.png) Here you can see that $B3$ and $B2$ have entirely different perceptions on the order of the two events, but that would break causality, so it cannot be true. What is wrong in my representation of relative systems of reference?


Amount of fluid that moves along a path

Let suppose that $\vec{F}$ is the velocity vector field of of a fluid in space, and $\vec{s}$ is a straight path of arbitrary length. Let's suppose that $\vec{F}$ and $\vec{s}$ are parallels and point in the same direction, so that $\vec{F} \cdot \vec{s} = F s$. How can we account for the total amount of fluid that moves along the path (per unit of time). **Related question**


Astronomical detection significance from magnitude error

At this website: http://heasarc.gsfc.nasa.gov/docs/swift/analysis/threads/uvot_thread_afterglows.html The passage at the bottom states that a V-band magnitude of 17.62, with an error $\pm$0.02 is a 49.4-$\sigma$ detection significance. How is this value calculated? Could you provide the working? I have a similar problem, with a set of magnitudes and errors, and would like to know at what magnitude limit I can claim a statistically significant (6-sigma) detection. Thanks


Selection rules in Stark effect

The energy level of an electron could be shifted by an electric field. $\langle n, l,m|[L_z,z]|n^{\prime},l^{\prime},m^{\prime}\rangle=(m-m^{\prime})\hbar \langle n, l,m|z|n^{\prime},l^{\prime},m^{\prime}\rangle$. Because $[L_z,z]=0$, we must have $m= m^{\prime}$ for non-vanishing $\langle n, l,m|z|n^{\prime},l^{\prime},m^{\prime}\rangle$. This is one selection rule in Stark effect. Then how to rigorously prove $l^{\prime}=l\pm1$ is the other selection rule for $\langle n, l,m|z|n^{\prime},l^{\prime},m^{\prime}\rangle\neq0$?


Mass-spring system on an incline

![block-spring system](http://i.stack.imgur.com/oCVEr.jpg) I am reviewing for an exam next week, and this is one of the questions I am stuck on. I have the mass-spring system above with spring constant $k$ on a frictionless incline. I would like to find the total energy of the system at any time $t$. I know that the total energy of the system is going to be the sum $E = K + U_g + U_s $ My first question is that I am confused about how to approach the problem. If I take my axes along the ramp, then the problem becomes one dimensional. My idea was set up my coordinate system like that so I could describe the displacement of the mass around it's equilibrium position on the ramp so I could find $K + U_s$, and then switch back to a more traditional coordinate system where I take my axes along the perpendicular sides of the ramp to find $U_g$. Does this approach work? If not what should I be doing?


What characterizes a metallic sound, and why do metals have a metallic sound?

We know that when we strike a metal, it usually has a characteristic "sharp" sound, unlike when we strike wood, say. What characterizes this "metallic sound"? Does it have a well-defined power spectrum? What are its generic properties? Also, why do metals have a metallic sound? My best guess would be that metals have high Young's / Bulk modulus and so the resonances for typically sized metals should be at fairly high frequencies, and since they are very elastic, they wouldn't dissipate energy all that easily, so the resonances would have small FWHM and hence be sharp. So my guess is that a "metallic sound" has distinct resonance peaks in its power spectrum, whereas for a piece of wood, for example, these peaks merge into each other to create a continuum.


Element of area in 4-dimensional space-time

How would you proof that $$ \mathrm {Tr} (\mathbf{S\cdot \bar S })=0$$ where $\mathbf S$ is an element of area delimited for the 4-vectors $\mathbf u$ and $\mathbf v$ given by $$S^{\alpha \beta}\equiv u^\alpha v^\beta-u^\beta v^\alpha$$ and $$\bar S^{\alpha \beta}\equiv \frac{1}{2}\epsilon^{\alpha \beta \gamma \delta} S_{\gamma \delta}$$ is the dual of $\mathbf S$. I used an analogy with the Maxwell field tensor $\mathbf T$. I know that $\mathrm {Tr} (\mathbf{T\cdot \bar T })=\frac{4}{c}\mathbf E \cdot \mathbf B$. Building the analog vectors $\mathbf E$ and $\mathbf B$ but with $\mathbf S$ I get that $ \mathrm {Tr} (\mathbf{S\cdot \bar S })=0$. But I'm looking for a more ilustrative solution to this problem. Any ideas?


What is the Hubbard-Holstein model?

Please explain as simply as possible what the Hubbard-Holtstein model is and what it is used for.


Einstein's box - unclear about Bohr's retort

I was reading a book on the history of Quantum Mechanics and I got intrigued by the _gendankenexperiment_ proposed by Einstein to Bohr at the 6th Solvay conference in 1930. For context, the thought experiment is a failed attempt by Einstein to disprove Heisenberg's Uncertainty Principle. ![Einstein's box as drawn by Bohr](http://upload.wikimedia.org/wikipedia/en/6/65/Ebohr4.gif) > Einstein considers a box (called Einstein's box; see figure) containing > electromagnetic radiation and a clock which controls the opening of a > shutter which covers a hole made in one of the walls of the box. The shutter > uncovers the hole for a time Δt which can be chosen arbitrarily. During the > opening, we are to suppose that a photon, from among those inside the box, > escapes through the hole. In this way a wave of limited spatial extension > has been created, following the explanation given above. In order to > challenge the indeterminacy relation between time and energy, it is > necessary to find a way to determine with adequate precision the energy that > the photon has brought with it. At this point, Einstein turns to his > celebrated relation between mass and energy of special relativity: $E = > mc^2$. From this it follows that knowledge of the mass of an object provides > a precise indication about its energy.   >  \--source Bohr's response was quite surprising: there was uncertainty in the time because the clock changed position in a gravitational field and thus it's rate could not be measured precisely. > Bohr showed that [...] the box would have to be suspended on a spring in the > middle of a gravitational field. [...] After the release of a photon, > weights could be added to the box to restore it to its original position and > this would allow us to determine the weight. [...] The inevitable > uncertainty of the position of the box translates into an uncertainty in the > position of the pointer and of the determination of weight and therefore of > energy. On the other hand, since the system is immersed in a gravitational > field which varies with the position, according to the principle of > equivalence the uncertainty in the position of the clock implies an > uncertainty with respect to its measurement of time and therefore of the > value of the interval Δt. **Question:** How can Bohr invoke a General Relativity concept when Quantum Mechanics is notoriously incompatible with it? Shouldn't HUP hold up with only the support of (relativistic) quantum mechanics? Clarifying a bit what my doubt is/was: I thought that HUP was intrinsic to QM, a derived principle from operator non-commutability. QM shouldn't need GR concepts to be self consistent. In other words - if GR did not exist, relativistic QM would be a perfectly happy theory. I was surprised it's not the case.


How much of a star falls into a black hole?

http://blogs.discovermagazine.com/badastronomy/2011/04/05/astronomers-may- have-witnessed-a-star-torn-apart-by-a-black-hole/ A lot of the star in the disc, a lot of the star in the jets, precisely how much of the star actually falls into the black hole?


How would you go about finding the natural frequencies (resonance frequency)

How would you go about finding the natural frequencies of solid materials like wood (e.g., teak, pine), stone (e.g., marble, granite) liquid, etc?


Interpretation of Free Damped Vibrations

I'm studying vibrations; so I'm using Beer-Johnston-Cornwell Dynamics book. I am worry about the equation for Underdamped Vibration, which in the book it is: $$x_{(t)}=x_0e^{-\lambda t}\sin(\omega_d+\phi)$$; where $$\omega_d=\sqrt{\omega_n^2+c^2/4m^2}$$. I think that $x_0$ would be replaced by the result vector of constants $c_1$ and $c_2$ affected by the factor $e^{-\lambda t}$. It means, an $x_m$ or an amplitud, but not the initial position, because it could be 0, with an inicial velocity. Also the graphic, it depicts the boundary equation with $x_0$. I attached a picture. Can you help me and comment? How I should interpret $x_0$? Maybe I misunderstood this topic. ![damp](http://i.stack.imgur.com/obyol.jpg)


What does "optical conductivity" mean?

Does it just mean "AC electric conductivity"? If so, why have a special name for it, and why mention optical specifically? The wikipedia page on it is very sparse. This (warning, PDF) document just says: > The term “optical conductivity” means the electrical conductivity in the > presence of an alternating electric field. Which sounds exactly like AC conductivity to me. Is it different? If not, then why the special name?


What Happens When A Gravitational Wave Interacts With Another One?

If two gravitational waves came in contact with each other what would happen? In another question entirely, what happens when a higher gravitational field interacts with a weaker one.


Determine the number of days with North-East wind direction from the number of days with North and East wind direction?

I have some data like: Wind flow from north direction = “X” Numbers of days. Wind flow from east direction = “Y” Numbers of days. Then is there any formula to know numbers of days wind flows from North-East direction?


Will larger balls make a Newton's cradle swing more stable?

I got a rather small Netwon's cradle and when I start it, the effect is not very good since all the balls start to swing where the effect we want is obvious. The balls are small and I wonder if larger balls would make the swing more ideal, like only the outer balls appear to move and the 3 inner balls should be still? I know product recommendations are off-topic, but I'm leaning towards buying a cradle with larger balls and perhaps you can advice which alternative to get that best replicates the original ideal cradle where only the outer balls move and the inner balls appear still?


Spin up, spin down and superposition

I'm just starting to study quantum mechanics. Please explain the error in this thinking: You set up decay of two $\pi$ mesons and get $2\mathrm{e}^-$ on Mars and $2\mathrm{e}^+$ on Earth. On Earth you may or may not measure the spin of those positrons, with 50% probability that they are the same spin. On Mars, your buddy "immediately afterwards" takes a $\mathrm{He}^{2+}$ ion and adds the electrons. If you made the measurement, the $\mathrm{He}$ has a 50%+ chance of being in a higher-then-base energy level due to Pauli exclusion. If you didn't make the measurement, the $\mathrm{He}$ has a much lower chance of being in a higher-than-base energy level. Your buddy measures the energy state. Result: You just superluminally transmitted $\approx 0.8$ bits of information.


What could cause a 30 k g acceleration?

I'm reading the specs of an IC (Cypress 5LP SoC) and it says it's available in 30 k g shock resistance configuration. The fastest acceleration I heard of so far was hitting a golf ball hard, which would be around 1000 g. Does anyone have an example of a mechanical impact of 30 000 g (300 000 m$\cdot$s$^2$), applicable to electronics devices?


A rod of length $L$ & mass $M$ is rotating in a circle about one end then calculate tension in the rod at a distance $x$ from the support

A rod of length L & mass M is rotating in a circle about one end then calculate tension in the rod at a distance 'x' from the support ? For its solution why should we take mass of L-x portion of rod instead of taking mass upto x distance from support as we have the formula T = m w2 x I m in confusion....


What's the advantage of NASA's flying saucer over traditional aerodynamic models?

NASA has recently tested a saucer-shaped spaceship. What's the advantage of this new design over traditional aerodynamic designs? The test launch was performed from within atmosphere which would offer higher air drag. What's the positive side of this that this big thing has been ignored?


$SU(2)$ gauge symmetry

Take the Lagrangian with one fermion: $$ \mathcal{L} = -\frac{1}{4}F^{\mu\nu}_aF^a_{\mu\nu} + \bar{\psi}(i\gamma^\mu D_\mu - m)\psi$$ where the gauge covariant derivative $D_\mu = \partial_\mu+i\frac{g}{2}t^aW^a_\mu$. The Lagrangian is invariant under a local $SU(2)$ transformation: $$ \psi(x) \rightarrow \exp \left[-i\theta^a(x)t^a \right]\psi(x) $$ $$W^a_\mu(x) \rightarrow W^a_\mu(x) +\frac{1}{g}\partial_\mu\theta^a(x) + \epsilon^{abc}\theta^b(x)W^c_\mu(x)$$ Often, we say that $W_\mu^a$ transforms according to the adjoint representation of $SU(2)$ but how can we say that based on the previous equation?


Does Hall Effect increases the resistance of a wire?

If a current in a wire is flowing perpendicular to a magnetic field, the Hall effect will be observed, which is caused by the forces from magnetic fields 'pushing' the electrons to one side or the wire. So, does it increases the resistance of the wire, as there is now less area for current to flow through due to one side of the wire being occupied by these (assumed)immobile electrons?


Radioactive stability of some nuclei

While studying radioactivity I found that even the most radioactive substances i.e substances with the shortest half lives do not completely degenerate. Suppose there is a 1 mole sample of an element **X** whose half life is **1 day**. But as the _degeneration equation_ is _exponential_ in nature even after 1 million years some amount of **X** would remain in the sample. What i want to know is how come certain **_atoms of an element possess such stability that they do not generate after a million years_** , **_while some atoms taken form the same sample degenerate after 1 day._** Is it because they have some different particles in their nucleus or something ? What is the role of **quantum tunnelling** ?


Why does an airplane have more lift near the ground?

I've noticed that an airplane appears to have more lift when it's almost touching the ground then it has 100 feet or more in the air. What causes this to occur?


An atomic bomb explodes inside of an "unbreakable" container which is on a scale. Does the "weight" of the container change?

This may or may not be an incredibly stupid thought experiment, but a short time ago I read that most of the "mass" in the proton was actually energy from the quarks and gluons, as opposed to the actual mass which was coupled to the Higgs field. This made me start thinking about objects on macroscopic scales and the effects of energy and mass in a closed macroscopic system. I have only a superficial highschool level understanding of relativistic mass, and I have read extremely vigorous debates and arguments about the significance of the mass-energy equivalence, and what it means in terms of the "weight" of an object. So my question is that if said atomic bomb were to explode inside of a completely impenetrable container of some sort, would the measured weight of the container change, assuming no radiation was able to escape the container itself?


Amount of thermal energy in the Earth?

Does anyone know the amount of thermal energy that the Earth's mantle and core possess? I don't mean the maximum limit of electrical power we could generate with geothermal plants, but rather: if you took the Earth and magically cooled it down so that its temperature became homogeneously 0K, what would the change in the Earth's internal energy be? (ignoring the Sun) If we don't have that data, does anyone have an idea how to perform an order of magnitude estimate?


Uncertainty relation and Energy-Position interference

How would you prove that the simultaneous measurements of position and energy are not subject to interference? I was thinking in calculate the commutation relation between $x$ and $H$ (Because $\Delta E=\Delta H$), but I realized that $[H,x]\neq0$, so I tried to use a more general expression for the Uncertainty Principle that says that if $H_1$ and $H_2$ are Hermitian operators then $\Delta H_1 \Delta H_2\geq\frac{1}{2}|\langle [H_1,H_2]\rangle|$, but again, $[H,x]\neq0$. Can you suggest me a way to do this? Thanks.


If you look at most published articles, or those on the preprint server, you find the vast majority of articles have more than one authors. Are collaborations needed to be productive in physics? In other fields like math, single author articles are the norm.


A wobbly pan on an induction cooker, is it less effective?

According to the Wikipedia, one of the limitations of the induction cooker is that the bottom surface of the pot should be flat. Accordingly, I commented on a question on Seasoned Advise, but I'd like to know whether this is actually true (My username is BaffledCook there). For a wok, it seems reasonable to believe the induction cooker will not be effective, but how about a wobbly pan? How wobbly should it be before it really affects efficiency?


Calculating size/weight of the base of a banner to withstand wind load

I want to build what is essentially a banner - a rectangular piece of fabric that will be mounted to a stand and displayed vertically. This object will be outdoors, so the most important factor in it staying upright is that it be able to withstand wind load. I know how to calculate wind load on an object, and I understand that the force of that wind load can be considered to act on the center of mass of the banner, but I'm not sure how to figure the size or weight of the base into the system to determine whether the object will fall over under the wind load. For example, if I were to make a banner 3 feet wide and 6 feet tall, With a base also 3 feet wide, and 3 feet deep, how much would the base need to weigh to withstand a 30 mph gust of wind? (Of course, the ideal answer would show me how to calculate this for myself, to try different scenarios)


How to find a hyperbolic escape orbit with a starting point and a desired velocity after sphere exit

How would you find an escape orbit which includes a given starting point (not assuming a circular starting orbit) and escapes the sphere of influence of the current body with a given remaining velocity? A problem perhaps similar to Lambert's Problem, but instead of an arrival location and time, it is a final velocity. I have a solution that someone suggested to me, that takes the final velocity as a hyperbolic excess velocity, uses that to calculate specific energy and semi major axis of the departure orbit. However they then assume that you're departure burn will occur at the periapsis of the new orbit and derive the eccentricity, and determine other elements of the departure orbit using that. The approach would work, though you have to use various guesses as the starting position until you find one that results in an orbit departing in the correct direction and the start is always at the periapsis of the departure orbit. I am hoping that there is an answer to this problem that doesn't require starting at the periapsis of the escape orbit, as I think that would broaden the range of potential departure directions from a given point. However I don't know how I would begin to calculate that.


Kac-Moody algebras in 5 dimensional Kaluza-Klein theory

I am trying to make sense to the issue of how does the Kac-Moody algebra encode the symmetries of the non-truncated theory. Let's contextualize a little bit. Ok, so in the 5 dimensional Kaluza-Klein we start with pure (5 dimensional) gravity. Then we assume a $M_4×S^1$ ground state where we have Minkowski space with a circle. The topology of this ground state breaks the 5D diffeomorphism invariance because now only periodic transformations are allowed. So considering a coordinate change $$x\to{}x'=x+\xi,$$ we can Fourier expand $$\xi(x,\theta)=\sum{}\xi(x)e^{in\theta}.$$ It is known that if we truncate at $n=0$ we get a theory of gravitation in 4 dimensional space-time, electromagnetism and a scalar field (sometimes called the Dilaton). So, the remnants of the initial 5 dimensional diffeo invariance are 4 dimensinal diffeo invariance, $U(1)$ gauge and a scale invariance for the dilaton. In the non truncated theory we will keep all the modes in the previous Fourier expansion and we will have more symmetries. The 4d diffeo and $U(1)$ gauge do survive in the non truncated version but not the scale invariance of the dilaton. So far so good. But, how to know the symmetries of the full theory? well, with Kac-Moody generalization of Poincaré algebras. For example in http://arxiv.org/abs/hep- th/9410046 says this. But HOW exactly? I would like a clear explanation of how this Kac-Moody algebra encodes the remnant symmetries of the original diffeo 5 and also how the zero modes of this algebra correspond to diffeo4 and $U(1)$ EDIT:: this is the algebra I am talking about $[P_{\mu}^{(n)},P_{\nu}^{(m)}]=0$ $[M_{\mu\nu}^{(m)},P_{\lambda}^{(n)}]=i(\eta_{\lambda\nu}P_{\mu}^{(m+n)}-\eta_{\lambda\mu}P_{\nu}^{(m+n)})$ $[M_{\mu\nu}^{(n)},M_{\rho\sigma}^{(m)}]=i(\eta_{\nu\rho}M_{\mu\sigma}^{(m+n)}+\eta_{\mu\sigma}M_{\nu\rho}^{(m+n)}-\eta_{\mu\rho}M_{\nu\sigma}^{(m+n)}-\eta_{\nu\sigma}M_{\mu\rho}^{(m+n)})$ $[Q^{(n)},Q^{(m)}]=(n-m)Q^{(n+m)}$ $[Q^{(n)},P^{(m)}_{\mu}]=-mP^{(n+m)}_{\mu}$ $[Q^{(n)},M^{(m)}_{\mu\nu}]=-mM^{(n+m)}_{\mu\nu}$


Problem with shell model and magnetic moment of Lithium-6

I have a problem with the calculus of magnetic moment of Li-6. The configuration of protons is $1p_{3/2}$, and the neutrons' one is the same. I have to add the magnetic moment of uncoupled proton and uncoupled neutron. I use the following formula for $J=l+\frac{1}{2}$ (J is the particle spin): $$ \frac{\mu}{\mu_N}=g_lJ+\frac{g_s-g_l}{2}$$ For the proton I have: $g_l=1; g_s=5.58 \rightarrow \frac{\mu}{\mu_N}=J+2.29=3.79$ For the neutron I have: $g_l=0; g_s=-3.82 \rightarrow \frac{\mu}{\mu_N}=-1.91$ So the total $\frac{\mu}{\mu_N}=3.79-1.91=1.88$, exactly 1 more than the correct value, 0.88! What's wrong?


Poles for a particle scattered in a delta potential

I am working on problem a professor gave me to get an idea for the research he does, and have hit a point where I'm having a difficult time seeing where I need to go from where I'm at. I would also like to go ahead and apologize for not knowing how to format correctly. I was given that a particle is scattered with the given Hamiltonian: $$ H = P^2 - g\delta(x) $$ Where $\delta(x)$ is the Dirac delta function. I was able to find the states in momentum representation, and was supposed to Fourier transform them to get the position representation states. Doing this leads to an integral with simple poles on the real axis, and can be solved by moving the poles above or below the real axis by some constant, applying the residue theorem, and taking the constant to zero. This leads to three different solutions, depending on if I move one pole up and one down (two possibilities), or both poles into a contour. While I was talking to my professor, he mentioned that the solutions only work for certain values of $x$, and that the range of $x$ is given by what makes the arc in the contour go to zero. I'm actually having trouble seeing this, since that introduces an ambiguity in the solutions. If I shift the left pole up and the right pole down and close the contour in the upper plane, this means that x has to be positive, in order to get a decaying exponential in the integral. However there is nothing stopping me from moving the poles in the opposite way, and closing above to obtain a different solution for $x>0$. Is there something wrong in the math, or is the way I move the poles governed by the physical situation I'm interested in? (Which would be no plane waves moving the left for $x>0$ d.) I should mention that I am assuming: $$ |\psi> = |p> \+ |\psi_{sc}> $$ Where p is the incoming momentum and $|\psi_{sc}>$ is the scattered portion of the wave function. Working in momentum representation, I obtain: $$ \psi_{sc}(k) = \frac{g - \frac{g^2 i}{2p+ g i}}{2\pi(k^2-p^2)} $$ Where k is the momentum variable, and p is the fixed momentum of the incoming particle. The transform I obtain is: $$ \psi_{sc}(x) = \eta \int_{-\infty}^{\infty} \frac{e^{i k x}}{k^2-p^2} dk $$ This is where I run into the problem with the poles. I know there shouldn't be any waves traveling to the left for $x>0$. My final goal is to check my states by checking the reflection and transmission coefficients and confirming they add up to 1.


black hole no-hair theorems vs. entropy and surface area

I was revisiting some old popular science books a while ago and two statements struck me as incompatible.   * No-hair theorems: a black hole is fully-described by just a few numbers (mass, spin etc) irrespective of the type or configuration of the matter/energy within the event horizon.   * Surface area measures entropy: you can't reduce the total entropy of the universe by throwing a box of hot gas into a black hole, its size/entropy will increase by the necessary amount. Suppose I have two black holes, A and B. They are identical and interchangeable, same mass, spin, charge... I have two boxes, BoxA and BoxB. BoxA contains a kilogram of salt in the form of a single crystal and BoxB contains a kilogram of salt as loose powder. **BoxA has less entropy than BoxB** but is otherwise identical. I throw BoxA into A and BoxB into B. The no-hair theorems seem to imply that A and B will increase in size by the same amount. The entropy theorems seem to imply that B will end up larger than A. What am I missing here?


Particles for all forces: how do they know where to go, and what to avoid?

Here's an intuitive problem which I can't get around, can someone please explain it? Consider a proton P and an electron E moving through the electromagnetic field (or other particles for other forces, same argument). They exert a force upon one another. In classical mechanics this is expressed as their contributing to the field and the field exerts a force back upon them in turn. In quantum mechanics the model is the exchange of a particle. Let's say one such particle X is emitted from P and heads towards E. In the basic scenario, E absorbs it and changes its momentum accordingly. Fine. How does X know where E is going to be by the time it arrives? What's to stop E dodging it, or having some other particle intercept X en route? Are P and E emitting a constant stream of force-carrying particles towards every other non-force-carrying particle in the universe? Doesn't this imply a vast amount of radiation all over the place? I am tempted to shrug of the entire particle exchange as a mere numerical convenience; a discretization of the Maxwell equations perhaps. I am reluctant to say "virtual particle" because I suspect that term means something different to what I think it means. Or is it a kind of observer effect: E "observes" X in the act of absorbing it, all non-intercepting paths have zero probability when the waveform collapses? Or have I missed the point entirely?


Is there a maximum amount of photons that can exist in a certain amount of space?

If you have a set amount of space, lets say 10 cubic centimeters, and you would be able to trap photons in there.   If you would then add more and more photons to that space, could you then go on infinitely or would you eventually run into a maximum amount of photons that can be in that space?


Observer effect, do this mean literally someone or just any interaction with other matter?

I am a layman and was wondering, the quantum observer effect. The regular notion to laymen seems to be literally "if you look at it", but as I am coming to understand the world I live in better I feel it means just coming in contact with something. Is this an ongoing question? Whether a particle that is not interacting with anything undergoes changes in state? Though now that confuses me too, If it is in two states at once. What denote say spin to the left from spin to the right, what denotes the origins of the calculations we make?


Self-adjoint and nonpositive differential operators

I recently tumbled over a statement in a geophysics paper (PDF here). They have a wave equation which they formulate as > $$ \frac{1}{v_0}\frac{\partial^2}{\partial t^2} \begin{pmatrix}p \\\ > r\end{pmatrix} = \begin{pmatrix} > 1+2\epsilon&\sqrt{1+2\delta}\\\\\sqrt{1+2\delta} &1\end{pmatrix} > \begin{pmatrix}G_{\bar x \bar x}+G_{\bar y \bar y}&0\\\0&G_{\bar z \bar z} > \end{pmatrix} \begin{pmatrix} p\\\r \end{pmatrix}\tag{20} $$ and they claim that > To achieve stability, the rotated differential operators $G_{\bar x \bar > x}$, $G_{\bar y \bar y}$, and $G_{\bar z \bar z}$ should be self-adjoint and > nonpositive definite as are the second-order derivative operators > ($\tfrac{\partial^2}{\partial x^2}$, $\tfrac{\partial^2}{\partial y^2}$ and > $\tfrac{\partial^2}{\partial z^2}$). (see eq. 14 and statement under eq. 20). They also claim that self-adjointness and nonpositiveness of the differential operators of the wave equation are necessary to conserve the energy in this system, and that if they were not self-adjoint, numerical instabilities occur. > We have solved the problem by introducing the self-adjointness to the > operator matrices in equation 20 to make sure that energy is conserved > during the wave propagation to avoid amplitude blowup in the modeling. In this case the wave equation consists of two coupled elliptical PDEs. What happens in general, when some operators are not bounded and linear? Unfortunately I don't have the mathematical background to understand this statement.


How to derive the Bethe stopping power formula

I need the derivation of Bethe formula for stopping power, but I can't see the corresponding paper to this matter. > Application of Ordinary Space-Time Concepts in Collision Problems and > Relation of Classical Theory to Born's Approximation. E. J. Williams. _Rev. > Mod. Phys._ **17** no. 2-3 (1945) pp. 217-226. Can anyone help me about this paper please?


What makes an equation an 'equation of motion'?

Every now and then, I find myself reading papers/text talking about how _this_ equation is a constraint but _that_ equation is an equation of motion which satisfies _this_ constraint. For example, in the Hamiltonian formulation of Maxwell's theory, Gauss' law $\nabla\cdot\mathbf{E}=0$ is a constraint, whereas $\partial_\mu F^{\mu\nu}=0$ is an equation of motion. But why then isn't $\partial_\mu j^\mu=0$, the charge-conservation/continuity equation, called an equation of motion. Instead it is just a 'conservation law'. Maybe first-order differentials aren't allowed to be equations of motion? Then what of the Dirac equation $(i\gamma^\mu\partial_\mu-m)\psi=0$? This is a first-order differential, isn't it? Or perhaps when there is an $i$, all bets are off... So, what counts as an equation of motion, and what doesn't? How can I tell if I am looking at a constraint? or some conservation law?


Vertical component of moving weight at a 45 degree angle

Here's an easier one. I use the leg press machine at the gym so I don't have to worrying about hurting myself while lifting heavier weight. The weight glides on a track that looks to be 45 degrees. What's the equation to figure out how much weight I would be able to squat normally. IE the vertical component of moving 400lbs at a 45 degree angle.


How to express $ds$?(when we know expression for $ds^{2}$)

We know that $$ds^2 = g_{\mu\nu}dx^{\mu}dx^{\nu},$$ Can you say how to calculate $ds$?


How to calculate the energy required to generate a vortex?

I am looking for the formula to calculate how much energy it requires to create a vortex. Can anyone assist please?


Problems book recommendation on supersymmetry, supergravity and superstring theory

I'm learning supersymmetry, supergravity and superstring. I want some problems books to have some idea in this area. Is there this kind of books? Or are there some papers that have many solved model?


Pulley problem with two masses

Assume a pulley and a massless rope. On one end there is a 10.0kg mass. On the other there is a 34.5kg mass. What is the acceleration of the 34.5kg mass? I created a free-body diagram with $m_1g_1$ pointing up and $m_2g_2$ pointing down, so the total tension is $m_1g_1 - m_2g_2$. This results in 240 N downwards, so I divide by 34.5kg to get $6.96 m/s^2$. However that is incorrect. What is my mistake?


Double Pendulum

The equations of motions for the double pendulum is given by $$\dot{\theta_1} = \frac{6}{ml^2}\frac{2p_{\theta1} - 3\cos(\theta_1 - \theta_2)p_{\theta2}}{16 - 9\cos^2(\theta_1 - \theta_2)}$$ and similarly for the other pendulum. In respect to what does the change in angle for the first pendulum refer to? Is it with respect to time? So that $\dot{\theta_1} = \frac{d\theta}{dt}$?


Historical background of wave function collapse

I wonder what were the main experiments that led people to develop the concept of wave function collapse? (I think I am correct in including the Born Rule within the general umbrella of the collapse paradigm.) Are there any instances where cases once thought to be examples of collapse have since been explained as the normal time-evolution of the wave function? EDIT: I'm going to have to make an objection to Ron Maimon's very excellent answer about particle tracks as evidence of collapse. I've been waiting for someone to suggest what I personally have always considered the prototype of the wave function collapse, namely the appearance of flecks of silver on a photographic plate when exposed to the light of a distant star. This has the essential elements of collapse in a way that ordinary photographic exposures do not. The mere appearance of dots on a photographic plate does not signal the collapse of anything: it is readily explainable as a consequence of the rate of silver-bromide reduction being proportional to light intensity. It is only when the intensity becomes so very low that the time taken to accumulate enough energy for a single conversion becomes unreasonable that we must consider the explanation of wave function collapse. The tracks in the cloud chamber do not demonstrate this phenomenon since the energy needed for the creation of the tracks is already available in the supersaturated gas. It is not necessary for the incoming particle to supply energy for the creation of the track, so there is no need to collapse its wave function. The straightness of the tracks is explained by Mott as an ordinary consequence of time-evolution of the wave function. There is no experimental proof that a single "particle" cannot be responsible for multiple tracks in the cloud chamber, because the tracks are not tagged according to which particle created them.


How to calculate the projected area at different angles/vectors?

Please help me with the following. I want to know if there is an equation/set of equations to find out the projected area of a (3-D) cube when it is oriented at different angles of attack to the fluid flow, rendering the velocity vector of the impinging fluid on the cube surface to change with each angle of attack. In particular, what is the projected area when the cube is oriented at such an angle that the flow velocity vector passes through one of the vertex and the geometric centre of the cube? Thanks.


direction of Pockels Effect. Refractive index eigendirections

There is a linear electro-optic effect called Pockels Effect The brief is that refractive index changes due to electric field. If there is an anisotropy (like birefringence) and electric field is in perpendicular to optical axis then refractive index changes for different polarization of ordinary rays. It gets eigendirections 45 degrees to electric field. Why so? Why it gets min 45 degrees but not exactly towards direction of E?


Isometry group from information about the center of the group

I am reading this paper on Dyons and Duality in $\mathcal{N}=4$ super- symmetric gauge theory. The author finds the zero modes or a dirac equation obtained by considering first order perturbations to the Bogolomony equation for bps monopoles. He finds out that when the simple root used for embedding the SU(2) monopole is simple then the bosonic zero modes which I believe is the solution for the higgs field transform as $1 \oplus 0$, and when it is not a simple root it transforms as two doublets. Hope this is not required for my question. **Then he says that the zero modes are SENSITIVE to the center of SU(2), and hence the isometry group is SU(2)? What does sensitive to the center mean? Is center the subset which commutates with all elements of SU(2)? How can it tell me about the isometry group?**


Evolution principle of the physical laws

I wanted to know if there is a physical theory that considers that the laws of physics undergo an evolutionary process. That see the law of physics or the absence of them, as something dynamic, and that with time they slowly converge to something we know today. A kind of simulated annealing of the physical laws.


Can Helium Disappear from Earth?

Can Helium disappear? As we know Helium is lighter than air, so basically Helium fly off from Earth. Is it possible that in the future we will run out Helium?


Where do electrical charges go, on a nonconductor dielectric when we make it charged?

My question is about electrically nonconductor dielectrics. We know such materials don't possess free charges.They have atoms bound together and every atom has specific numbers of electrons turning around its nucleus. When we make a dielectric charged: where does this charge go? what keeps the charge fixed on the structure? Are they bound to specific atoms? If your answer is "yes",so What makes that atom specific to possess the charge? (same questions about positively charging.)


Applications to the Van der Pol equation?

What are some applications to the Van der Pol equation? Are there any physical examples?


Understanding Quantum Physics

I have very little background in physics, and none in quantum physics, but I've been reading about how sub-atomic particles behave probabilistically, so I was wondering, is it possible (even though the probability would be unimaginably small) that all the particles which make up my body are located somewhere 100 light years away? Or have I misunderstood the concept?


Domain wall and kink solutions from solitions equations

A general solition equation can be obtaion from scalar field theory $$\varphi(x) = v\tanh\Bigl(\tfrac{1}{2}m(x - x_0)\Bigr),\tag{92.6}$$ where $x_0$ is a constant of integration when we drived this solution from Lagrangian. **Comparing to the sine-Gordon theorywith this article , can we convert our equation to the sine-Gordon equation?** Because this equation represented for 2 dimensions (1 space dimension and 1 time dimension) then how will I get Kink solutions and domain wall equation from the above equation. Any elaboration would be helpful for me. This is a continuing post of Solving the soliton equation without energy. **ADDITION: If I want to visualize the solition equation (92.6) by graph then what would be the best process?**


Does Light Ever Cease Once Created

I was reading an article on Huffington Post (link to article) about the Multiverse theory. In the article it said this: > The big news last week came from the Background Imaging of Cosmic > Extragalactic Polarization 2 (BICEP2) experiment at the South Pole, which > saw imprints in the cosmic microwave background— **the oldest light in the > universe** , dating from shortly after the big bang... I am far from scientific, actually just a web dev, but I have an interest in learning and reading about science, so please explain your answer in a way I can understand it. My question is in regards to the bold type in the quote. Does light ever actually end, go away, or just cease to exist once it is created? Will this "oldest light in the universe" ever completely cease to exist, or does light stay around indefinitely.


What are the prerequisites to study topological quantum computation/topological phases of the matter?

I am an undergraduate student and I would like to approach the subject of topological order with focus on topological quantum computation, I know (very) little QFT and basic algebraic topology (if that matters...). To give a reference frame, I'd like to understand papers like http://arxiv.org/abs/quant-ph/9707021v1 or Preskill's great presentations: http://online.kitp.ucsb.edu/online/exotic_c04/preskill/, and later to take it as my field of research for a PhD. So my questions are: 1) What kind of studies are useful/needed for this field? I assume Quantum Computation/Information are, but what else? Quantum Field Theories, Topology, Quantum Optics...? 2) Which are univerities or research centers active in the field (preferably in Europe)? 3) Any reference suggested? (Better if beginner-focused)


Friction influencing the motion of a mass and non-inertial frame of reference

So... recently a Newtonian-mechanics related exercise has raised in me many questions basically ragarding which forces influence a given system and how, I know they might sound bit too "noobish" but I just had to get them out of my chest (yes it really gave me something to think about in the last day). perhaps you should take a look at the above mentioned exercise to better understand what I am trying to figure out ![image of the system](http://s21.postimg.org/ek3mmimw7/Nuova_immagine_bitmap.png) So, I'm trying to figure out how this system works... the exercise says (please forgive me for all the grammatical errors you will eventually find): > # Problem > > a cube with a mass $\ M=50Kg$ is standing on a plain surface and can move on > it without friction. On that cube is standing another cube with a mass $\ > m=10Kg$, at a given distance $\ d=0.5 m$ from the upper-left corner of the > cube. Initially, when everything is stationary, a force $\ F=100N$ is > applied to the bigger cube horizontally (I assume that the force $\ F$ is > constant); at the moment $\ t=2s$ the smaller cube falls. Calculate the > friction coefficient between the two cubes. # Variables used: > $\ μ$= friction coefficient > > $\ g$= gravitational acceleration on earth =$\ 9.81 m/s^2$ > > $\ a_M$= acceleration of the bigger cube ($\ M$) > > $\ a_m$= acceleration of the smaller cube ($\ m$) > > $\ F_f$= friction force So, I assume that the bigger cube carries the smaller one which moves with a constant, negative acceleration until it falls (please correct me if I'm wrong), with equation of motion $\ x(t)= x_0 + v_0t + 1/2 at^2$ so $\ 0.5m = - 1/2 at^2$ ------> $\ a= -0.25 m/s^2 $ now... $\ F- μ mg = Ma_M$ in other terms (and again please correct me if I'm wrong): $\ F$ is partially dissipated by the friction between the cubes, and the bigger cube moves under the influence of a force which equals to the applied force $\ F$ minus the friction force $\ F_f= μ mg$ now the universe collapses: the solution says: > $\ μmg = ma_m$ > > $\ a_r = a_M - a_m = [ μg (m + M) -F] / M$ which leads to $\ μ=0.15 $ But... what leads to those last two equations?


How can I derive this Hamiltonian?

I have a Lagrangian $L$, a momentum $p$ and a Hamiltonian $H$: $$L=\frac m 2(\dot z + A\omega\cos\omega t)^2 - \frac k 2 z^2$$ $$p=m\dot z + mA\omega\cos\omega t$$ $$H=p\dot z - L=\frac m 2 \dot z^2 - \frac m 2 (A\omega\cos\omega t)^2 - \frac k 2z^2$$ And I want to calculate $\frac {\partial H} {\partial z}$ and $\frac {\partial H} {\partial p}$. I understand from this question that I need to algebraically manipulate $H$ to express it in terms of $p$ and $z$. The answers there suggested trying to express $\dot z$ in terms of $z$ and $p$, and presumably I need to express $t$ in terms of $z$ and $p$ as well. But that seems like its going to lead into very nasty territory... first of all, I've got quadratics, so that's not an invertible function. Second of all, for $t$, if I have to use inverse trigonometric functions, then it'll only be valid over a particular range of the variable. Could I get some pointers on how to tackle this calculation? Is inverting the various relations really the way to go?


About the geodesics in general relativity

I'm learning general relativity from the book " Einstein's General Theory of Relativity - Øyvind Grøn and Sigbjorn Hervik". The field equations are derived by the Hilbert - Einstein action and are written in the form $$ R_{\mu\nu}-\frac{1}{2}R\,g_{\mu\nu}+\Lambda\,g_{\mu\nu}=kT_{\mu\nu}$$   where $k$ is acostant that should be determined. The authors find this $k=\frac{8\pi G}{c^4}$ by imposing the well known conditions of "Newtonian Limit". The first requirement is the following > The particles in free fall moving along geodesics induced by $g$. This is a postulate? By the Einstein equivalence principle we know that locally we can eliminate the effects of gravity and so living in a (flat) Minkowskian space-time but there is not mention of the above statement. I agree on the fact that once we have a curved space-time, in presence of gravitational fields, it is natural to require that the free falling particles move along the geodesics, but this can be proved or it is postulated?


Question about Bose-Hubbard Model

By using an optical lattice, how can one change the interaction term $U$? And how is the superfluid phase achieved in the hard-core boson regime? Why are these phases identified as superfluid or mott insulator?


Step by step algorithm to solve Einstein's equations

I cannot completely understand what is a _regular_ method to solve Einstein's equations in GR when there are no handy hints like spherical symmetry or time- independence. E.g. how can one derive Schwarzschild metric starting from arbitrary coordinates $x^0, x^1, x^2, x^3$? I don't even understand the stress-energy tensor form in such a case - obviously it sould be proportional to $\delta(x - x_0(s))$, where $x_0(s)$ is a parametrized particle's world-line, but if the metric is unknown _in advance_ how do I get $x_0(s)$ without any a priory assumptions?


Transmutation with cosmic radiation possible?

Can cosmic radiation (alpha radiation) transmute the material of a space craft, particular carbon, titanium and aluminum? Where can i find transmutation tables or formulas to calculate the possibility of a transmutation and the outcome?


How many seconds is a temporal meter?

Is there a proof that time is a 4th dimension? If it is, then why not measure it in units of the previous three? Logical right? How many seconds is a temporal meter?


When to use ideal gas law in fluid mechanics?

The ideal gas law (aka the equation of state) is given by $$ p/\rho_N = k_BT, $$ where $\rho_N$ is number density. When am I allowed to use this to describe a fluid?


A few questions about the Fermi Level/Energy

My first question is, how is the Fermi Energy for a material actually determined? I know this derivation, but it seems to say that the Fermi Energy is just based on the electron density (and maybe some effective mass) of the material. Is that really all that determines it? Secondly, I'm trying to figure out how the interfaces of various materials work in terms of their bands, but it's not clear to me exactly what _must_ be true in all cases (vs what is often but not necessarily true, or what is theoretically but rarely practically true). For example, Anderson's Rule starts by aligning the "vacuum levels" of the two materials, but then this article says that it's not a great idea to use the vacuum level, and the Anderson's Rule article says it's just not that accurate a rule, anyway. Similarly, it seems like the Schottky-Mott Rule isn't very successful either. Additionally, I've read somewhere that the Fermi Level (the electrochemical potential, the sum of the chemical potential and electric potential) has to be continuous everywhere in both of the materials, so that results in the _chemical potentials_ (i.e., the $T \neq 0$ Fermi Energies, which were normally different in the two materials) lining up, and that happens by having an _electric potential_ difference across them. But this picture from wikipedia then seems to suggest that either what I just said is wrong, or the label should really be "Fermi _energy_ " (or chemical potential) in their definitions. Which is it? So, what can I always depend on and know is true in these situations? Thank you!


Peierls Argument for Absence of Long Range Order

I'm really confused about the argument in Cardy's book for why there can't be long range order in 1D for discrete models. Let me just copy it out, and hopefully someone can explain it to me. He takes an Ising-like system as an example. We start with the ground state with all spins up, and we want to see if this state is stable against flipping the spins in some chain of length $l$. This chain has two domain walls at the endpoints, so we get an energy change of $4J$. Then the claim is that there is an entropy of $\log l$ associated with this chain, since "each wall may occupy $O(l)$ positions." If this were true, we would get a free energy change of $4J-\beta^{-1} \log l$, and this would imply that the ground state is unstable to flipping very long chains. The only part I'm not on board with is the claim about the entropy. I would say that if $L$ is the length of the system, then we have $L$ places to put the chain, so we get an entropy of $\log L$. Certainly as $L\to \infty$ this gives no long range order, as expected. So, is the entropy $\log l$ or $\log L$? (Incidentally, I'm perfectly happy with his argument in 2D...)


List of the basic quantum mechanical variables

Is there a list of basic quantum variables/attributes that all quantum particles have? Ex. An electron has charge, position, speed, momentum, etc. Is there a complete list of these variables? I would figure not all quantum particles share the same set of variables? A photon has position and an electron has position but a photon does not have a charge and an electron does, though I guess it is said a photon has neutral charge.


How to calculate Temperature Humidity Wind Index?

I would like to know how to calculate Temperature Humidity Wind Index (THW Index)? I know how to calculate Heat Index and Wind Chill. I am asking this because my weather station Davis Vantage Pro2 calculates THW index but I could not find any information on NOAA. This is what it says under Help for my weather station: > The THW Index uses humidity, temperature and wind to calculate an apparent > temperature that incorporates the cooling effects of wind on our perception > of temperature.


Why are magnetic lines of force invisible?

We can very well feel the magnetic field around a magnet, but we can't see it. Why is that so? Also, can we cut a portion of the field and use it?


Kinetic energy of a photon and Schwarzschild radius

I have read here, that $\frac{1}{2}mv^2$ must not be applied on a photon ever. If i want to calculate escape velocity $v_e$ i need to use $\frac{1}{2}mv^2$ because we say that kinetic energy _(positive)_ must be same or larger than gravitational potential _(which is negative)_ in order for an object to escape. It is done like this: $$ \begin{split} W_k + W_p &= 0\\\ \frac{1}{2}mv_e^2 + \left(-\frac{GMm}{r}\right) &= 0\\\ v_e &= \sqrt{\frac{2GM}{r}} \end{split} $$ Than we say if there is a black hole inside certain radius we call Schwarzschield radius $r=R_{sch}$ not even light can escape because its escape velocity is smaller than needed to escape. At the border of the sphere with $R_{sch}$ light can barely escape, so it must hold that $c$ equals escape velocity $v_e$. So we write down the equation below and derive $R_{sch}$. $$ \begin{split} c &= \sqrt{\frac{2GM}{R_{sch}}}\\\ R_{sch} &= \frac{2GM}{c^2} \end{split} $$ This is a well known equation, but it is derived allso using $\frac{1}{2}mv^2$ for light (photons). This is in contradiction with 1st statement in this post. So is the last equation even valid???


What exactly is an anti-neutrino?

According to the the definition of anti-particles, they are particles with same mass but opposite charge. Neutrinos by definition have no charge. So, how can it have an anti-particle?


Should you run when under rain?

When it's raining, would you get less wet if you run or more wet? I think you will get less wet, because rain is coming down in a constant volume and if you run you will have a shorter amount of time you will get less volume I'm not too sure though, some people say when you run you are running into a more volume of rain because rain is not all coming up to down, it will hit you more horizontally


The role of the affine connection the geodesic equation

I apologise in advance that my knowledge of differential geometry and GR is very limited. In general relativity the equation of motion for a particle moving only under the influence of gravity is given by the geodesic equation: $$ \ddot{x}^\lambda + \Gamma^\lambda_{\mu\nu}\dot{x}^\mu\dot{x}^\nu =0. $$ I am looking for a conceptual description of the role of the affine connection, $\Gamma^\lambda_{\mu\nu}$ in this equation. I understand that it is something to do with notion of a straight line in curved space. Comparing it to the equation for a free particle according to Newtonian gravity: $$ \ddot{x}_i = -\nabla\Phi, $$ Then it kind of looks like the affine connection is our equivalent of how to differentiate, except that our scalar field is now some kind of velocity?


Bullet entering target

I've been given this question: > A bullet entering a target with an initial velocity of $ u $ loses $\frac > {u}{n}$ of its velocity after penetrating a distance $ a $ into the target; > how much further will it penetrate? My question is, doesn't some information about the nature of the bullet's acceleration need to given? How otherwise would I formulate the kinematic equations?


Is the energy of a photon continuous/discrete?

I was struggling today with this question: does a free photon have a continuous energy spectra? Free means in no context of any energy system (eg. an atom, em field). Although I'm asking myself if the quantization of the electromagnetic field is omnipresent and will always make the energy discrete? Edit: This leads me also to the question: if we have 2 energy levels (like in hydrogen: ground state and first excited) the uncertainty principle tells us, that the energy isn't quite exact defined: $\Delta E\Delta t \ge \hbar$ . Therefore the final energy of the emitted photon won't have a discrete energy, since it would be sth. like $E_{photon} = E_{0} + \Delta E$ ?!


Where does a spinning figure skater's energy go when she slows down?

Today in physics class we were talking about angular momentum and rotational kinetic energy. My teacher used the classic example of a figure skater spinning on ice - when she pulls her arms in, her angular momentum is conserved and her angular velocity increases, meaning that her rotational kinetic energy also increases. Of course, this increase in energy must come from somewhere - in this case, it comes from the figure skater doing work on her arms and pulling them in toward her body. Then I started wondering - if the figure skater slows her rotation by extending her arms, she decreases her rotational KE. Where is her energy going? Or to put it another way, what force is doing work on the figure skater in order to decrease her energy?


What is Maupertuis' principle good for?

The strength of Hamilton's principle is obvious to me and I see the advantage. Now, for conservative systems we also have Maupertuis' principle that says: $$ \delta \int p dq =0$$ and I am not sure how to derive an equation of motion from this? Is this of any use in practical computations? So, can one apply this principle for example to the harmonic oscillator?- I have never seen anybody using it. Further, I read in Goldstein's classical Mechanics that the variation in Maupertuis' principle is not the one in Hamilton's principle, since we have constant Hamiltonian and changing time, whereas Hamilton's principle has constant time and varying Hamiltonian (in general). I am a little bit wondering about this, since you could easily get Maupertuis' principle from Hamilton's principle: $$ \delta \int L dt = \delta \int p \dot{q} - H dt = \delta \int p \dot{q} dt = \delta \int p dq =0,$$ if $H$ is constant. Can anybody here explain to me, why we have to use a different variation and how one can use this principle?


Band-limited double-step Fresnel diffraction for images with radio waves?

Is it possible to use band-limited double-step Fresnel diffraction to assemble a holographic image with radio waves? If not is there a simular principle?


Question on expression in "J.S.Bell : On the Einstein Podolsky Rosen paradox"

I have a question on the article `J. S. Bell, On the Einstein Podolsky Rosen paradox, Physics 1, 195, 1964.` (link) My question concerns the expression (3) of the article, at page 196. I don't understand what is the reasoning that leads to this expression of the expectation value... I think I miss something but I don't know what. This is what I understood from now on : $\vec{\sigma_1}$ and $\vec{\sigma_2}$ are the spins of the two particles that move apart and must be exactly opposite according to quantum mechanics when measured in a direction of the component $\vec{a}$. First, did I understand well and do we really have $$A(\vec{a},\lambda) = \vec{\sigma_1}.\vec{a} = \pm 1 \\\ B(\vec{b},\lambda) = \vec{\sigma_2}.\vec{b} = \pm 1$$ then ? If not, what does $A(\vec{a},\lambda)$ and $B(\vec{b},\lambda)$ correspond to ? A sort of $sign$ function or something like in the next section? Secondly, why $$ <\vec{\sigma_1}.\vec{a}\; \vec{\sigma_2}.\vec{b}> = -\vec{a}.\vec{b}$$ Is it because $\vec{\sigma_1}$ and $\vec{\sigma_2}$ are opposite ? Thanks !


Asymptotic Completeness, generalized free fields, and the relationship of thermodynamics with infinity

Asymptotic completeness is a strong constraint on quantum field theories that rules out generalized free fields, which otherwise satisfy the Wightman axioms. If we were to take a limit of a list of continuous mass distributions $\rho_n(k^2)$ that approaches a distribution in some topological sense, however, is there anywhere an analysis of how the behavior of the $\rho_n(k^2)$ would approach the behavior of the free field? The following statement seems too bald (from the review "Outline of axiomatic relativistic quantum field theory" by R F Streater, Rep. Prog. Phys. 38, 771-846 (1975)): "If the Källén-Lehmann weight function is continuous, there are no particles associated with the corresponding generalized free field; the interpretation in terms of unstable particles is not adequate". Surely as we take the support of a generalized Lorentz invariant free field to be arbitrarily small we could expect that the behavior, at least as characterized by the VEVs, which constitute complete knowledge of a Wightman field, would eventually be arbitrarily close to the behavior we would expect from a free field? Classical thermodynamics has a complicated relationship with infinity, in that the analytic behavior of phase transitions does not emerge unless we take an infinite number of particles, but the behavior of very large numbers of particles nonetheless can approximate thermodynamic behavior rather well. By this elementary analogy, it seems premature to rule out generalized free fields. It also seems telling, although weakly, that the Källén-Lehmann weight function of an interacting field is nontrivial in quasiparticle approaches. Being able to derive an S-matrix requires that a theory must be asymptotically complete, however real measurements are always at finite time-like separation from state preparations, with the interaction presumably not adiabatically switched off at the times of the preparation and measurement, so that something less than analytically perfect asymptotic completeness ought to be adequate. EDIT: To make this more concrete, the imaginary component of the mass $1$ propagator in real space at time-like separation is $I(t)=\frac{J_1(t)}{8\pi t}$. If we take a smooth unit weight mass distribution $$w_\beta(m)=\frac{\exp(-\frac{\beta}{m}-\beta m)}{2m^2K_1(2\beta)}\ \mathrm{for}\ m>0,\ \mathrm{zero\ for}\ m\le 0,$$ for large $\beta$ this weight function is concentrated near $m=1$, with maximum value $\sqrt{\frac{\beta}{\pi}}$. For this weight function, the imaginary component of the propagator in real space at time-like separation is (using Gradshteyn&Ryzhik 6.635.3) $$I_\beta(t)=\int\limits_0^\infty w_\beta(m)\frac{mJ_1(mt)}{8\pi t}\mathrm{d}m= \frac{J_1\left(\sqrt{2\beta(\sqrt{\beta^2+t^2}-\beta)}\right) K_1\left(\sqrt{2\beta(\sqrt{\beta^2+t^2}+\beta)}\right)}{8\pi tK_1(2\beta)}.$$ Asymptotically, this expression decreases faster than any polynomial for large $t$ (because the weight function is smooth), which is completely different from the asymptotic behavior of $I(t)$, $-\frac{\cos(t+\pi/4)}{4\sqrt{2\pi^3t^3}}$, however by choosing $\beta$ very large, we can ensure that $I_\beta(t)$ is close to $I(t)$ out to a large time- like separation that is approximately proportional to $\sqrt{\beta}$. Graphing $I(t)$ and $I_\beta(t)$ near $t=1000$, for example, and for $\beta=2^{20},2^{21},2^{22},2^{23},2^{24}$, we obtain ![Graph of $I\(t\)$ and $I_\\beta\(t\)$ near $t=1000$](http://i.stack.imgur.com/Pk3pG.png) $I(t)$ and $I_\beta(t)$ are very closely in phase, as seen here, until $t$ is of the order of $\beta^{2/3}$ in wavelength units. We can take $\beta$ to be such that this approximation is very close out to billions of years (for which, taking an inverse mass of $10^{-15}m$, $\sqrt{\beta}\approx \frac{10^{25}m}{10^{-15}m}=10^{40}$), or to whatever distance is necessary not to be in conflict with experiment (perhaps more _or_ less than $10^{40}$). This is of course quite finely tuned, however something on the order of the age of the universe would seem necessary for what is essentially a stability parameter, and the alternative is to take the remarkably idealized distribution-equivalent choice $\beta=\infty$ as usual. ~~[I would like to be able to give the real component of the propagator at time-like and space-like separations for this weight function, however Gradshteyn &Ryzhik does not offer the necessary integrals, and nor does my version of Maple.]~~ EDIT(2): Turns out that by transforming Gradshteyn&Ryzhik 6.653.2 we obtain $$R_\beta(r)\\!=\\!\int\limits_0^\infty\\!\\!w_\beta(m)\frac{mK_1(mr)}{4\pi^2 r}\mathrm{d}m= \frac{K_1\left(\\!\sqrt{2\beta(\beta-\sqrt{\beta^2-r^2})}\right) K_1\left(\\!\sqrt{2\beta(\beta+\sqrt{\beta^2-r^2})}\right)}{4\pi^2 rK_1(2\beta)},$$ which _is_ real valued for $r>\beta$. As for $I_\beta(t)$, the approximation to the mass $1$ propagator at space-like separation $r$, $R(r)=\frac{K_1(r)}{4\pi^2 r}$, is close for $r$ less than approximately $\sqrt{\beta}$. For the real component at time-like separation, ~~it is almost certain that~~ one ~~simply~~ replaces the Bessel function $J_1(...)$ by $Y_1(...)$.


Physical interpretation of different selfadjoint extensions

Given a symmetric (densely defined) operator in a Hilbert space, there might be quite a lot of selfadjoint extensions to it. This might be the case for a Schrödinger operator with a "bad" potential. There is a "smallest" one (Friedrichs) and a largest one (Krein), and all others are in some sense in between. Considering the corresponding Schrödinger equations, to each of these extensions there is a (completely different) unitary group solving it. My question is: what is the physical meaning of these extensions? How do you distinguish between the different unitary groups? Is there one which is physically "relevant"? Why is the Friedrichs extension chosen so often?


How can I add an acceleration vector to a velocity with a different direction?

I am in my last year of high school and am struggling with some homework. I'm sorry if this question is incredibly stupid, but I simply can't find the answer in my notes. If I have an object with a velocity of 10ms-1 travelling at a bearing of 090 degrees (to the right) with no acceleration, and then all of a sudden it spontaneously gains an acceleration of 2ms-2 at a bearing of 180 degrees (downwards), how do I calculate the velocity (magnitude and direction) of the object for every second after the object gains this acceleration? Thank you very much for your help, I really want to be able to understand this!


How are symmetries defined mathematically?

I have started working on differential geometry very recently. I am little bit familiar with mathematical concepts such as manifolds, differential forms and associated concepts. As I was speeding through I caught interest in Lie groups and whose invaluable connection with the symmetries. I would be happy if anybody answers to these few questions.   1. I want to know what is a symmetry from the mathematical point of view?    2. How are $su2$, $su3$, and other symmetries first formulated?    3. What was the motivation?    4. How do we identify internal symmetries and corresponding conserved physical quantities?


What is the relative speed of two near-light speed particles headed towards each other?

I understand that nothing can move faster than light due to time dilation. I want to build upon my understanding of Einstein's theory of Special Relativity, so I came up with this hypothetical problem for myself: > If both particles are moving at 0.9c towards each other (according to an > observer on Earth), what speed does each particle have relative to the > other? I'm stumped since I am only familiar with Newtonian classical mechanics to solve this (which would be wrong). I am aware of Lorentz' factor $\gamma =\frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}$ and the time dilation equation $t_m = \frac{t_o}{\sqrt{1 - \frac{v^2}{c^2}}}$, but I'm not sure how to apply it in this instance. I'm just looking for a gentle push in the right direction. Edit: My prior research:   1. This article, https://what-if.xkcd.com/1/, talks about the effect of a baseball thrown at 0.9c would have as it traveled towards a batter. This is not what I'm looking for since it doesn't explain the math behind finding the collision speed between two high velocity entities.   2. This question on Yahoo, https://answers.yahoo.com/question/index?qid=20110529201519AAxbxvm, asks a similar question to mine, but no math is involved to demonstrate the calculations behind the reasoning.   3. This question on Physics.SE, Collision between a photon and a massive particle, does not answer my question either. I have not been able to find a similar Physics.SE question which can provide me to clues to my own hypothetical question.    4. I have also Googled the following phrases: > i. "collision faster than the speed of light" > > ii. "collision of high speed particles" > > iii. "lorentz transformation in a collision" > > iv. "collision analysis at light speed"


Spontaneous symmetry breaking: How can the vacuum be infinitly degenerate?

In classical field theories, it is with no difficulty to imagine a system to have a continuum of ground states, but how can this be in the quantum case?   Suppose a continuous symmetry with charge $Q$ is spontaneously broken, that would means $Q|0\rangle\ne0$, and hence the symmetry transformation transforms continuously $|0\rangle$ into anther vacuum, but **how can a separable Hilbert space have a continuum of vacuums deferent from each other?**      I saw somewhere that says the quantum states are built upon one vacuum, and others simply doesn't belong to it, what does this mean? and **then how could $Q$ be a well defined operator** which acting on a state (the vacuum) actually gives a state (another "vacuum") out of the space considered?


Angular momentum of 2d harmonic oscillator

So, I have the problem of determining the spectrum of H and L, in terms of creation and annihilation operators of angular momentum... The problem goes along with what is happening on this page. However, my professor is asking us for the commutators of H and L with a quantity A, which is defined as $$A=\hat a_x^2+\hat a_y^2$$ When I do this out, writing H as $1+\hat a_x^{\dagger}\hat a_x+\hat a_y^{\dagger}\hat a_y$, I find that $$[H,A]=\hbar\omega(a_x^3+a_y^3)$$ I have no problem with this, except that when I want to construct an eigenket representation of $E_{nm}$in terms of the $\hat a_L^{\dagger}\text{ and } \hat a_R^{\dagger}$ and $A^{\dagger}$, I am not sure how to use this result to find this. I know that the result should be $E_{n,m}=\hbar\omega[2n+1+|m|]$ Thanks


What the heck is negative effective mass?

I am reading this book:Solid State Electronic Devices by Ben G Streetman and Sanjay Kumar Banerjee.   I have some doubts in the article **3.2.2** Effective mass.   In this the aythors say that $E=\dfrac{1}{2}mv^2=\dfrac{1}{2}\dfrac{P^2}{m}=\dfrac{{\hbar} ^2}{2m}k^2$. >   * Electrons usually have thermal velocity of order $10^7$m/s. So Shouldn't > we use $E=mc^2$ rather than $E=\frac{1}{2}mv^2$. > The author further says that electron mass is related to curvature of (E,k) relationship as $\dfrac{d^2E}{dk^2}=\dfrac{\hbar^2}{m}\tag{3.2(d)}$. then the author says that the effective mass is given by $m^{*}={{\hbar^2}/{{\dfrac{d^2E}{dk^2}}}}\tag{3.3}$ >   * Is equation $3.3$ derived from equation $3.2(b)$. In equation $3.2(d)$ > we have $m$ the original mass and in eqn $3.3$ we have $m^{*}$ the effective > mass which is completely different from $m$. > In the end of the article the author says that the total force on the electron is given by $F_{tot}=ma$. After this the author says the external force applied on an electron is related to effective mass as: $F_{ext}=m^{*}a$. >   * From where this equation comes? Sometimes the effective mass $m^{*}$ is > negative then the equation implies that if we apply certain external force > on electrons in the crystal they will accelerate in the opposite direction, > how this is possible? What's going on? >


What breaks the symmetry between the electromagnetic and weak nuclear force?

I know the electromagnetic force is mediated by a photon and the weak nuclear force is mediated by two massive bosons. Are there any other insights into why the masses are so different?


Virtual particles/quantum tunneling - conservation of energy?

I'm confused as to how the above phenomena can take place since arent they breaking the law of conservation of energy (even, if temporarily)?


Question on Type HO/HE string theory

The Heterotic string state is a tensoring of the bosonic string left-moving state and the Type II string right-moving state. Therefore, I expect the spectrum to be: $$\begin{array}{*{20}{c}} \hline & {{\rm{Sector}}}&{{\rm{Spectrum}}}&{{\rm{Massless fields}}}& \\\ \hline & {{\rm{Bosonic}} - {\rm{R}}}&{{\bf{1}}{{\bf{6}}_v} \otimes {{\bf{8}}_s} = {{\bf{8}}_v} \otimes {{\bf{8}}_v} \otimes {{\bf{8}}_s}}&?\\\ \hline & {{\rm{Bosonic}} - {\rm{NS}}}&{{{\bf{8}}_v} \otimes {{\bf{8}}_v} \otimes {{\bf{8}}_v}}&?& \hline \end{array}$$   1. However, how does one calculate the massless fields of the Type I string theory using the spectrum of the type I string theory?   2. Furthermore, to calculate the mass spectrum of the Heterotic string, does one simply add the number operator of the bosonic string to that of the type II string and the same for the normal ordering constant? i.e. is it true that $$\begin{array}{l} m = \sqrt {\frac{{2\pi T}}{{{c_0}}}\left( {B + {{\tilde N}_{II}} - {a_B} - {{\tilde a}_{II}}} \right)} \\\ {\rm{ }} = \sqrt {\frac{{2\pi T}}{{{c_0}}}\left( {B + {{\tilde N}_{II}} - 1 - {{\tilde a}_{II}}} \right)} \end{array}$$ **Edit: I found the answer to Question 2. Check the answers section. I have answered my own question.**


Relationship between Connection and Material Derivative

Suppose $D\subset \Bbb R^3$ contains a fluid and that $f : D\times \mathbb{R}\to \mathbb{R}$ is a time dependent function defined on the fluid region. In that case, the material derivative is defined by $$\dfrac{D}{Dt}f(a, t) = \dfrac{\partial f}{\partial t}(a, t) + (\mathbf{u}\cdot \nabla)f(a, t)$$ Where $\mathbf{u}\cdot \nabla$ is the operator defined on scalar function $g$ by $$(\mathbf{u}\cdot \nabla )g = \mathbf{u}\cdot (\nabla g) = D_{\mathbf{u}} g$$ That is the directional derivative along $\mathbf{u}$ of the function $g$. On vector fields it is defined componentwise, that is, if $\mathbf{v} = (v_1,v_2,v_3)$ then $$(\mathbf{u}\cdot \nabla)\mathbf{v} = ((\mathbf{u}\cdot \nabla)v_1, (\mathbf{u}\cdot \nabla)v_2, (\mathbf{u}\cdot \nabla)v_3) = (D_{\mathbf{u}}v_1, D_{\mathbf{u}}v_2, D_{\mathbf{u}}v_3)$$ But that latter thing is clearly the Covariant Derivative of $\mathbf{v}$ along $\mathbf{u}$ when we consider the Levi-Civita Connection on $\mathbb{R}^3$ with the usual flat metric tensor, that is $$(\mathbf{u}\cdot \nabla)\mathbf{v} = \nabla_{\mathbf{u}}\mathbf{v}$$ Now, is this conclusion right? Can we really write the material derivative as $$\dfrac{D}{Dt}\mathbf{v}(a, t) = \dfrac{\partial \mathbf{v}}{\partial t}(a, t) + \nabla_{\mathbf{u}}\mathbf{v}(a, t)$$ and if it's right is there some usefullness in this relationship? I mean, I don't know that much of connections and how they can be used on Physics, but I know they are usefull. In that case, writing the material derivative in terms of a connection gives some advantage? Would it make sense if $\nabla$ were another connection other than the Levi-Civita connection?


From where does the force/energy of action-reaction comes from? (Newton's Third Law of Motion)

I was wondering, from where does the opposite force and thus energy comes from when we apply force. For example, lets say there are two persons (P1 and P2) on the universe, and no force is applied to them. P1 applies a force to P2. P2 will start to move. P1 lost some energy to do that. Now, according to the third law of newton, P1 will also get a force from P2 as a reaction. But from who does the energy of reaction comes from? I mean, I found it hard to believe that P2 is the one who losses energy, just for the sake of reaction. So, from where does this force, and thus energy, comes from? I am not an expert, I just know the basics. So please, be tolerant :P


How to derive the spin-orbit term in Kane Mele model?

The Kane mele model is a famous quantum spin hall model on honeycomb lattice (C.L. Kane and E.J. Mele, Phys. Rev. Lett. 95, 226801 (2005)). The Hamiltonian is $$H = - t\sum\limits_{\left\langle {i,j} \right\rangle \alpha } {c_{i\alpha }^\dagger {c_{j\alpha }}} + i{\lambda _{SO}}\sum\limits_{\left\langle {\left\langle {i,j} \right\rangle } \right\rangle \alpha \beta } {{\nu _{ij}}c_{i\alpha }^\dagger \sigma _{\alpha \beta }^z{c_{j\beta }}} $$ The first term is just the nearest hopping term for graphene, while the second term represents the spin-orbit interaction, which seems quite peculiar to me.   1. Must it be in the z direction? Why?   2. Why does the spin-orbit term only have the second nearest hopping term, but no nearest hopping term? How does it relate to the symmetry of honeycomb lattice?   3. Is there an intuitive way to understand this term? **Can somebody tell me the derivation for the second term?**


Black hole entangled with the cosmological horizon

Maldacena and Susskind recently proposed a interesting and very suggestive duality between entanglement and topological identification: http://arxiv.org/abs/1306.0533 But are such ideas applicable to horizons in general? Can I entangle a black hole horizon, with, say, a Rindler horizon as seen by an accelerated observer? Can I entangle a black hole with our cosmological horizon? Are these in principle, physically possible operations? **Note:** this question might be part of a multi-part question


How do I prevent shattering of glass?

I am not an expert of physics, instead I am more good at chemistry. I just wanted to ask that how do I prevent shattering of glasses on sudden large temperature changes? Sometimes, when I have to cool hot test tubes rapidly, the test tubes break down when I place them in water for cooling. Is there any way to prevent it?


About units and plural form

The plural form of physical units are always confusing me. I asked some senior students, some of them said we need to use plural form of units but some said units are always singular. For example, 1 meter is 1 meter. But it is 3 times, should it be 3 meters? I know that if we use the abbreviation form, no plural form, i.e. 3m. But if we use 'meter', for the case like 0.04 meter, should it be 0.04 meter or 0.04 meters? I think we should use 0.04 meter because it is less than or equal to 1. The last question I have is about foot and feet, when the magnitude is less than or equal to 1, I should use foot, otherwise, use feet instead, is that right?


How does an electron microscope work?

I am a physics novice. Google tells me that electron microscopes work much like their optical counterparts -- but the analogy falls apart for me when I think about what I'm "viewing." Obviously, you can _see_ light through the lenses, but what is the "image" analog for electron microscopes? Is it at-all like spraying an invisible shape with bullets and examining where collisions took place? Like if you shot at an invisible car with a tommy gun and were able to make out bullet holes -- so that the more bullets you shoot the better your image? And, just for completeness, I suspect this implies that the best resolution you can get is the bullet-size, or in this case the size of the electron. How do you map "objects" or whatever they are considered on that scale if they are smaller than an electron? Is our perception of how _small_ we can see limited by this cap?


Electron/Photon Scattering

Hey guys, I have a final tomorrow and I am going over some assignments. One of the questions from the assignment was: > A photon having 37 keV scatters from a free electron at rest. What is the > maximum energy that the electron can obtain? Now, I got the right answer (4.67 KeV) but I can't seem to figure it out now. Maybe I am just too tired. Any hints to where I should begin?


superconductor levitating in earth's magnetic field?

> **Possible Duplicate:**   >  Can superconducting magnets fly (or repel the earth's core)? I've seen superconductors levitating on magnets. But is it possible for superconductors to levitate on Earth from Earth's magnetic field?


Is there a stable numerical algorithm for FWHM that isn't 2.35*sigma?

This is a question that should have a simple answer, but which I can find no proper discussion of in the literature or on the internet. I start from the assumption that I have a noisy numerical signal with a peak in it - for example amplitude as a function of $x$, i.e. $A(x)$ - and I wish to determine its full-width at half maximum (FWHM). I am aware of two basic algorithms that may be used on numerical data:   1. Assume it's Gaussian, determine the r.m.s. value, and multiply by 2.35 ( $2\sqrt{2 \log{2}} $).   2. Determine the FWHM using a peak-finding algorithm that locates the peak $A_{max} = A(x_{peak}$), then locates the _first_ positions either side where $A(x)$ falls to $1/2 A_{max}$. Method 1 is numerically robust as you don't have to bin (i.e. smooth) the data that determines $A(x)$. However, if the peak is significantly non-Gaussian in shape you get a systematically wrong answer. Method 2 is direct, but there are plenty of numerical cases where there are multiple points in $x$ either side of the peak where $A(x)$ crosses $1/2 A_{max}$. And of course the number of crossing points depends sensitively on whether I smooth A(x) or not. My question is therefore whether there is a mathematically justifiable algorithm for determining FWHM numerically that doesn't assume the peak is Gaussian,


Plotting a wave function that represents a particle

The problem is this: > A particle is represented by the wave function $\psi = > e^{-(x-x_{0})^2/2\alpha}\sin kx$. Plot the wave function $\psi$ and the > probability distribution $|\psi(x)|^2$. This the problem 2.1 in the book _Fundamental University Physics Volume III_ by Marcelo Alonso and Edward Finn. The thing is I don't know what values $k, \alpha$ and $x_{0}$ should have. Probably I don't know what $\psi$ really represents in this case.


Basic mechanics problems, unsolvable by brute-force numerical integration

I'm looking for simple problems in theoretical mechanics that are impossible or unreasonably difficult to solve by means of "brute-force" numerical integration of Newton or Euler-lagrange equations. I'm interested in these beacuse I noticed that kind of "computer-nihilism" point of view is getting popular (at least among some students): A person says, that " _in the end we anyway doing real stuff by computer simulation_. _And for the numerical values of parameters we are usually able to numerically obtain the result with a given precision. So we just need to know how to write down the equations_ ".   And, therefore, " _there is no need to learn all that complicated stuff in theoretical mechanics_ ". Apart from obvious counter-arguments for this, I'd like to show that there are basic problems you are unable to solve without "the complicated stuff".   Let me give an example of such a problem:   **Given:**   1. A center, that creates some strange field with the potential $U(r)=-\frac{\alpha}{r^3}$. (Mysterious planet)   2. A body with mass $m$ scattering off this center. (Our space ship.)   3. A radius R, at which we want to stay as long as possible. **Find:** the impact parameter $\rho$ and the energy $E_0$ for our body, so it will stay in the "ring" $R<r<2R$ for as long as possible. The problem is easily formulated. And it is easy to solve even for "newbies" in theoretical mechanics. The specific feature of the problem -- there is no reasonable way of solving it by doing straightforward computer simulation. Can you propose other examples of problems with these properties?


How can one calculate the phase difference between two quantum harmonic oscillator (Hermite-Gauss) states?

The analytic solutions of a quantum harmonic oscillator are given by Hermite- Gauss states, which differ in the order $n$ of the Hermite polynomials. If two such states are plotted, there will be a relative phase shift. How can that be calculated analytically?


Is my electric power cord creating magnetic field when coiled?

Simple question. But I was wondering, does my mac power cord create a magnetic field when its all coiled?


Hydrogen cloud at the universe's beginning?

What prevented all of the hydrogen at the universe's start from coalescing into one gigantic star?


Trouble justifying calculation of force on a sphere due to radiation pressure

I know very little about electromagnetic waves and light in general so you are going to have to bear with me. I am attempting to calculate the force on a sphere in a plane wave of light where the light has intensity I. My calculations and research have concluded that, regardless of whether or not the sphere absorbs or reflects the light, that the force can be modeled as: $\frac{\pi{r}^2{I}}{c}$. This pdf document outlines what I am looking for (with $I$ replaced with $\langle{S}\rangle$) but in equation 24 it seems like a factor of $2\pi\text{r}$ appears out of nowhere. Another solution I have tried is a double integral in spherical coordinates and exploiting symmetry to find that $\text{Force} = \iint\limits_A \\! p \,\,{dA} $ where p is a function describing the pressure on an area element. Every time I calculate this pressure I find a function of one angle rather than of two angles. I find that my integral looks like: $$\int_{0}^{\pi}\\!\int_{0}^{\frac{\pi}{2}}\\!\frac{4\pi{r}^2{I}}{{c}}cos(\theta)^3sin(\theta) \, d\theta d\phi$$ which evaluates to the correct expression for force (up to the sign) but I have trouble justifying that the pressure does not depend on $\phi$ and only depends on $\theta$. In this instance, I am integrating over half of a hemisphere where $\theta$ is the polar angle and $\phi$ is the azimuthal angle or as described in this image. I feel like these calculations are incorrect and that the double integral over ${dA}$ should include an element reflecting the influence of $\phi$ as I cannot see any reason why the calculation would be independant of it.


If electromagnetic waves have magnetic fields, why beam of flashlight is not disrupted by a Magnet?

Wikipedia article about Electromagnetic Radiation says "As an electromagnetic wave, it has both electric and magnetic field components". And this discussion also confirms Light is EM wave. Since we know that light travels through electromagnetic waves. Does that mean when we hold a Magnet close to the beam of a flashlight, it should interact with the waves and cause funny things to happen ?


Do composite particles that are electrically neutral but have charged constituents radiate?

For example an electron radiates when accelerated. So does a positron. But is the radiation emitted by accelerated positronium the sum of the radiation emitted by each separately? If not, why not? If so, does this provide a way of testing whether a given neutral particle is composite? For example, does a neutron bremsstrahlung when decelerated?


polynomial fit and uncertainty

I am trying to fit a polynomial function by least square method and I want to know the uncertainty on my fitting parameters. In which book can I find a clear answer?


Fundamental Constants in a theory of everything (TOE)

Do physicists ever expect to be able to derive the fundamental constants of nature from theory? For example, if string theory or some other theory unites the four forces, would the theory be considered complete if it relies on these measured constants, or would a true theory of everything (TOE) require that these constants come out of the theory itself?


"Delayed choice" quantum imaging experiment - why wouldn't it work?

I was just reading a Nature news article (Entangled photons make a picture from a paradox, 27 August 2014) about a way to form an image of an object out of photons that have not interacted with the object, but which are entangled with photons that have. The article has a photograph of the image, which is of a cardboard cutout of a cat. If I understand correctly, the technique is that   1. a photon beam is split, e.g. via a semi-silvered mirror, resulting in two beams, call them A and B;    2. in each beam there is some device that turns each photon into two entangled photons. At this point there are four beams, call them A1, B1, A2 and B2.    3. The photons in beam A1 interact with the cardboard cat   4. The photons in beam A1 are allowed to interact with the photons from beam B1 in some fashion, _and then discarded_   5. The photons in beams B2 and A2 (neither of which has interacted with a photon that's interacted with the object) are combined somehow and used to produce the image. This is pretty neat - the article talks about using to image an object with very low-energy photons, while still being able to make an image on a screen using high-energy ones. However, the following occurred to me while reading about it: presumably, as with all experiments of this kind, it shouldn't in principle make any difference if we change the length of the beams. What would happen if we keep the length of beams A2 and B2 short, but we make beams A1 and B1 really long, say a few light hours? In this thought experiment the cardboard cat, along with the device for combining the photons from beams A1 and B1, is on Pluto, and the rest of the equipment is in a lab on Earth. It seems as if it should work just as well in this case as it does with everything in the same lab. But now we're seeing the image of the cat five hours before the photons interact with it - and by swapping the cat out for a different object, someone on Pluto could send us a message from the future. Presumably this can't really work, since it would be possible to construct a paradox if it did. But where is the issue? Have I misunderstood something about the setup (entirely possible since I'm getting this from the news article rather than the actual paper), or is there some fundamental reason why it would stop working if the A1/B1 beam length was much longer than the A2/B2 one? This seems similar to delayed choice quantum eraser experiments. However (if I remember correctly from looking into them years ago) in those experiments you have to combine all the beams back again to make the image in the end, so there's no way to construct a paradox. In this case the Nature news article seems quite clear that the photons are thrown away after interacting with the cardboard cat, and the image is produced _only_ from those that haven't - this seems to be a major difference.


Dirac Delta function

Although I find it mathematically dubious, we said that $\Delta \frac{1}{r} = -4\pi \delta(r)$. Now, I was wondering is there a similar relation to the delta function if we look at $\Delta \frac{1}{r^n} =?$.


Is the viscous sublayer size universal?

We know the non-dimensionalized distance of the viscous sub-layer is 5 wall units for a turbulent boundary layer. It is common to see this in Fluid Mechanics books but seems somewhat arbitrary. I want to know if this true for all conditions or does it depend on the Reynolds number (high viscosity, high speed, etc).


A little bit of background information: I'm planning to write a little booklet or web page about CPU/computer architecture, basically for my own education, because we didn't cover it in depth in college. I feel like I should be learning a lot more about the fundamentals about how computers work, if I want to be a better programmer. The idea is to present the workings of a CPU by describing it from the ground up. Starting at simple electrical circuits, then semiconductors, transistors etc. From my school physics class I remember that I was very often left with a lot of gaps about how different concepts are linked together. I want to avoid this. So here's my question: How does an electrical field really work? How come that an electron can exert a force on another electron without physical contact? What is it in an electron that creates the field, where does the energy for doing the work come from?


Are a measured object always part of the theory?

Is there a notion of measurement, which doesn't correspond to a yes/no question or with the idea of the comparison of two real world objects, which produces a real number? And does at least one of the object which are compared to produce the measurement result have to be modeled within the theory? (One of my motivations for that question is that I wonder if a human can make a measurement involving light without a length measurement and it also seems to me that almost all measurements involve looking at data, which therefore effectively makes them length measurements. And then I speculate if there are even other mearuements than these. For example, I can't test the quantum wave values unless I collapse it - its reality is hidden (this question is certainly not only about QM though.) and therefore the specific configuration is not real is a strict sense. Similarly there is no time mearuesment, is there? The configuration of the clock is just infered by a mearuement of light.)