Recommendations for Statistical Mechanics book

I saw Book recommendations   No reference to Statistical Mechanics there. I learned thermodynamics and the basics of statistical mechanics but I'd like to sit through a good advanced book/books. Mainly I just want it to be thorough and to include all the math.   And of course it's always good to give as much intuition about the material. Some things I'd be happy if it includes (but again it mostly just needs to be a good clear book even if it doesn't contain these) are:   1. As much justifications for the postulates if possible, I'm very interested in reading more about how Liouville's theorem connects to the postulates   2. Have examples of calculating partition functions, hopefully not just the partition function for the ideal gas. Thank you


Statistical mechanics prerequisite

I`ll be taking senior level and graduate course in statistical mechanics in a month. I was wondering what would be the best statistics and probability textbook to prepare for it? I`m currently reviewing classical mechanics and thermodynamics. I forgot quite a bit. Should I focus on a book that teaches mainly probability or both statistics and probability? I'm getting flags for this being a possible duplicate. I already have the statistical mechanics book I'll be studying from. But it doesn't have a decent enough review of statistics and probability which is what I'm trying to prepare for.


Blocking electro-magnetic waves between two near field antennas

Its been a while since I studied physics, so pardon my ignorance. I have two near field antennas. The range of the antennas is about 2-3 cm. I am looking for a way where I can block and unblock the communication between them. One of the way that I tried is to place a flat tightly woven wire and if I short the two ends of the wire, I was able to block the communication. My assumption is that it is effectively changing the frequency of the signal and thus blocking the signal. Is that what is happening? However, if the radio signal has good strength, the coil is not doing a good job. Is there any other better solution that I can achieve blocking/unblocking with electrical/electronics (without mechanical parts)?


Can the distance of a quasar be determined accurately?

As noted in A doubt about the age of the universe, the wiki about quasars still contains the following misleading sentence: "The highest redshift quasar known (as of June 2011) is ULAS_J1120+0641, with a redshift of 7.085, which corresponds to a proper distance of approximately 29 billion light-years from Earth." But even if the "strange 29 billion" is replaced by the "correct 12.9 billion", the fact remains that the actual measurement is "a redshift of 7.085". The "proper distance" is only a different way to express that measurement. It's not clear to me how "accurately" this describes the distance of the quasar, because the quasar surrounds a black hole and rotates quite fast. So there are at least two additional sources for the redshift, but how significant is their contribution?


Relativistic Lorentz force law

If we consider the the relativistic Lorentz force law: $$\frac{d}{dt} (m\gamma \vec{u})=e(\vec{E}+\vec{u} \times \vec{B})$$ How can we deduce: $$\frac{d}{dt} (m\gamma c^2)=e \vec{E} \cdot \vec{u}$$ Clearly dotting with $\vec{u}$ will give us the RHS. Which leaves us: $$\vec{u} \cdot \frac{d}{dt} (m\gamma \vec{u})=e \vec{u} \cdot \vec{E}$$ Could anyone help explain how to proceed and if this is the correct method? EDIT: If it helps: with reference to these notes i'm working through: http://www.maths.ox.ac.uk/system/files/coursematerial/2012/2393/8/WoodhouseLectures.pdf Page 86, eq (178), the paragraph underneath states 'The ﬁrst equation (which follows from the second)', this is what i'm trying to prove (a warning, the notes are riddled with errors..).


Can temperature be defined as propensity to transmit thermal energy?

I was recently surprised to learn that defining temperature isn't easy. For a long time, it was defined operationally: how much does a thermometer expand. Also surprising, temperature isn't a measure of thermal energy: 0 C water has more thermal energy than 0 C ice, but the same temperature. I came up with this definition, and I'd like to know if it's correct: Temperature is the propensity of something to give off heat, that is transmit thermal energy. Equivalently, it is the resistance of something to receive heat. Is that correct? Please justify your answer.


Trouble with the Lorentz law of force: Incompatibility with special relativity and momentum conservation?

In Physical Review Letters, there was a paper recently published: > Masud Mansuripur, _Trouble with the Lorentz Law of Force: Incompatibility > with Special Relativity and Momentum Conservation,_ Phys. Rev. Lett. 108 > (2012) 193901, arXiv:1205.0096. The paper considers the case of a stationary point charge at a fixed distance from a stationary magnetic dipole in some frame, stating that the magnetic dipole neither experiences a torque nor force from the stationary charge. Upon transforming to another frame, they argue that the magnetic dipole is transformed into a magnetic dipole and electrical dipole. This combination will therefore experience both an electrical force, and magnetic torque from the field of the moving charge. The authors therefore claim that there is a contradiction and therefore a problem with the consistency of the Lorentz force and special relativity. Is the paper correct, or has the author overlooked something?


Finding the Electric Field (and other information, besides)

The problem I am working on is: > Two parallel plates having charges of equal magnitude but opposite sign are > separated by 29.0 cm. Each plate has a surface charge density of 33.0 nC/m2. > A proton is released from rest at the positive plate. > > (a) Determine the magnitude of the electric field between the plates from > the charge density. > > (b) Determine the potential difference between the plates. > > (c) Determine the kinetic energy of the proton when it reaches the negative > plate. > > (d) Determine the speed of the proton just before it strikes the negative > plate. > > (e) Determine the acceleration of the proton. > > (f) Determine the force on the proton. > > (g) From the force, find the magnitude of the electric field. So far, I've only attempted to solve part (a). For part (a), seeing as they didn't provide the dimensions of the plates, I assumed them to be infinite-sized; furthermore, they both have to have the same area (although, I can't justify why this is so). So, for plate 1, which we'll consider positive: $\sigma = 33.0 \cdot 10^-9 C/m^2$ Since they don't say otherwise, that the plates do not have uniform charge distribution, we'll assume that the charge is in fact uniformly distributed; this is reasonable, seeing as they give one surface charge density for the entire plate. $\sigma = \frac{dq}{dA} \rightarrow dq = \sigma \cdot dA \rightarrow \int dq = \sigma \int dA \rightarrow q_1 = \sigma \cdot a^2$, $q_1$ being the charge of the entire plate. A similar process can be done for the second plate, which we'll regard as negative: $q_2 = \sigma \cdot a^2$ Hence, $q_1 = - q_2$ This is the only way of starting that I could think of; and as you may see, it wasn't very helpful. How should I approach this problem? * * * As per Michael Brown's suggestion, should I generate a Gaussian plate, infinite in size, and place it between the two sheets? Then, find the magnitude of the electric field from sheet, and multiply it by two?


Energy formula for separating $O_2$ from mixture of $O_2$, $NH_3$ and $H_2O$

I have a physics problem I'd like to make sure I get correct. The practical aspect of this problem is that the photosynthetic efficiency of algae is inhibited with dissolved O2 in the growth medium, and the economics of any algae cultivation system are very sensitive to photosynthetic efficiency. The problem I'm trying to solve is determine the theoretic energy to separate a gas mixture of, O2, NH3 and H2O (in boiling point order) into two gas streams, one of which is 99.99% pure O2 but the other stream can have as much as 50% of the original O2 along with NH3 and H2O. The input and output temperatures and pressures are STP and the input, at least, is at saturated humidity. Presumably, the energy is derived from the equations in this paper. Any approximate ideas of what the formula would look like? PS: The detailed model of the algae system is here.


A terminological question about work and energy

Work is force applied over distance. Is it also reasonable to say that work is (the same thing as) the transfer of energy? When work is done, the equivalent energy is transferred. But if energy is transferred, such as by heating something, is that (or could that be) called work? Also, if a black body is subjected to a stream of energy, which it absorbs and radiates away, is any work done? The black body, let's assume, remains at a constant temperature, but energy is transferred to it and then radiated away. Is any work done? So what I'm really asking about is how the term "work" should be used.


How to use Ising Spin model for prediction of time series "Phase"

I am investigating a 2d Ising Spin Lattice. I have been able to generate a Monte Carlo app that gives me the changing spin matrix through my iterations - like the many examples on the web. However, I am trying to predict an independent variable (atmospheric pressure). I believe it is possible to measure the pseudo "Magnetism" and "Energy" of the timeseries using the Ising approach - but I am not sure how to do that. I assume it entails flipping the "spin" based on the timeseries changes, but I am not really sure weather that is the right approach. If anyone could give me some advice that would be great. EDIT 1 I have a series of pressure readings:                Xt :1..N      Instead of using Metropolis flip where               deltaE = 2 * J * Si * sumOf(Sneighbours)     if (deltaE < 0 || exp(-deltaE/TEMP) > rand())            Si+1 = -Si      I need to set               Si+1 = someFunctionOf(Xt)      But I don't know what "someFunctionOf" should be. If I did I could update the Lattice from Xt, and workout a "live" Magnetism and I am hoping that might tell something about the current phase of pressure changes.


Quantum Collapse

When we observe a quantum object does it collapse into a point? Or does it collapse into a smaller wave of area that is out of our range of accuracy? My gut tells me the latter.


(Canonical) Partition function - what assumption is at work here?

The canonical partition function is defined as $$Z=\sum_{s}e^{-\beta E_s}$$ with the sum being over all states of the system. The way I saw this derived was by assuming that for each state, the probability of the system occupying that state is proportional to the Boltzmann factor: $P(E=E_s) = c \cdot e^{-\beta E_s}$. Summing the probabilities to one gives $Z=\frac{1}{c}$. My question is : What principles were used in order to get this probability distribution for the energy?


How does angular momentum transfer between a planet and its moon?

Could you explain how a moon draws angular momentum from a planet? I know that the gravitational force transfers momentum, but I don't understand the mechanics behind it.


Is forward scattering = no scattering?

What is forward scattering? If it is equivalent to no scattering, then why not call it "no scattering"?


The mathematical physics of saltatory neuron action potential transmission

Some time ago, I tried to understand saltatory electrical potential transmission on a myelinated axon but have not found a clear mathematical model of this mechanism. I understand the cable theory, but that seems only explains the normal continuous transmission within the cable either with or without insulation, instead of the saltatory one between two un-myelinated synapsis of an axon. What is the model that describes the interior and exterior interaction that results in the SALTATORY transmission mechanism? It would be even better if the answer to my question could be accompanied with reference to pertinent papers on the mathematical model on the saltatory transmission. Here is a reference on neuronal transmission modeled as electrical potential transmitted in a cable and as an application of the cable theory. My problem with this theory is that it does not account for the saltatory transmission mode of the action potential in a myelinated neuron.


What are the physics behind the Coriolis effect?

What causes the Coriolis effect?


Do Gauge Theories (CFTs) Have Phase Transitions as the 't Hooft Coupling is Varied?

With an eye toward AdS/CFT, I'm wondering if large $N$ CFTs have a (quantum) phase transition as the 't Hooft coupling is varied. To be more specific -- if I look at correlation functions of low-dimension, single trace operators as a function of $\lambda$, are they analytic, continuous, discontinuous in $\lambda$? How much do we know / any references? I'm already aware of 0811.3001.


Do Killing spinors know global information?

The conformal Killing spinor equations on $R\times S^3$ in Minkowski signature are \begin{equation} \nabla_\mu \epsilon=\pm \frac{i}{2}\gamma_\mu\gamma^0\gamma^5\epsilon \end{equation} whose solution is \begin{equation} \epsilon=e^{ix^0\gamma^5/2}\epsilon_0 \end{equation} where $\epsilon_0$ is a constant spinor. Note that we identify $S^3$ with $SU(2)$ and use the left-invariant vector fields of $SU(2)$ as an orthonormal frame. For more detail, please see around Eq.2.20 in the following paper. http://arxiv.org/abs/hep-th/0605163v3 When the superconformal index is interpreted as a partition function on $S^1\times S^3$ in the Euclidean signature, the conformal Killing spinors are modified by the Wick rotation $\epsilon =e^{-x^0\gamma^5/2}\epsilon_0$ as in the following papers. http://arxiv.org/pdf/1104.4482v3 http://arxiv.org/abs/1104.4470 However, $\epsilon =e^{-x^0\gamma^5/2}\epsilon_0$ is not well-defined on the temporal circle $S^1$. This issue also happens on the Killing spinor on $AdS_3$. The metric of $AdS_3$ in the Minkowski signature can be written as \begin{equation} ds^2=-\cosh^2 \rho dt^2 + \sinh^2\rho d\phi^2+d\rho^2~. \end{equation} The Killing spinors on this coordinate takes the form \begin{equation} e^{\frac 12\rho\gamma_3}e^{-\frac {i }2 (\phi+ t)\gamma_1}\epsilon_0 \end{equation} as in http://arxiv.org/abs/hep-th/9310194 . However, when you consider the elliptic genus $Tr(-1)^Fq^{L_{0}-c/24} \overline{q}^{\overline L_{0} -\overline c/24} y^{J}$, the bulk duals which satisfy the E.O.M of the supergravity theory are the Euclidean thermal $AdS_3$ and the family of its $SL(2,Z)$ transformations such as the BTZ BH. Then, again the Killing spinors on the Euclidean thermal $AdS_3$ take the form \begin{equation} e^{\frac 12\rho\gamma_3}e^{(-\frac {i\phi }2 + \frac{t}{2})\gamma_1}\epsilon_0 \end{equation} which are also not well-defnined on the temporal circle. The Killing spinor equations themselves are local, so does the Killing spinor know only local information? If so, how you distinguish the Killing spinors in the NS from in the R boundary condition on 2-torus?


Pure mathematical exposition vs A "for physicists" approach: Which is better?

I'm an undergrad interested in theoretical physics. Don't know enough to decide which specific subfield I may go in, but particle theory (strings, QFT, GR) and mathematically heavy condensed matter seem appealing. I have taken quite a few pure math courses in my undergrad career so far (algebra, analysis, topology etc), but now that I've exhausted most of the undergraduate curriculum, its time to start some advanced topics such as Algebraic Topology and Lie Algebras etc. My concern however is that these courses are offered by the mathematics department and thus the exposition and focus is aimed towards training mathematicians rather than someone who may find them useful in another field. So as expected in such a class, the main goal is to understand and prove the main theorems of these fields. However, as a physicist I'm not sure to what extent I can benefit from such an exposition. It would be highly time consuming to take such an approach to mathematics for someone to which mathematics is not the ends itself, but rather the means. On the other hand, there is the option of using "math for physics" type books such as Nakahara or Frankel to learn these topics, which would be a lot quicker and would have a physicist as a reader in mind, but no doubt, depth of understanding would be lost with such an approach. So my question is: Is it worth it for an aspiring theoretical physicist to learn advanced mathematical topics from graduate math classes (and as a result sacrificing the time spent physics), or would it be a better use of his time to focus on the physics and pick up the math from a text or a class of the "for physicists" type? How would the answer change if applied to a mathematical as opposed to theoretical physicist? Note that over here I'm talking about topics such as Algebraic Topology, Differential Geometry, Representation Theory etc.


Ground state energy of hydrogen molecule ion

In this paper, it is mentioned: > Furthermore, since the energy of $H_2^+$in the ground state must be lower > than that of an H atom in the ground state,the negative (attractive) forces > in the $H^+ – H$ interaction must play the dominant role How do we intuitively expect the ground energy of hydrogen molecule ion $H_2^+$ should be less than that of hydrogen atom without doing exact quantum mechanical calculation.


How (or why) equivalence principle led to Einstein field equations?

If equivalence principle was origin of general relativity what was the process that this principle led Einstein to developed his theory of general relativity?


Can current and resistance result in establishing voltage?

I want to know whether current & resistance can cause a voltage across the circuit, according to Ohm's Law $V=IR$. Is it possible to use a low current to produce a high voltage?! How can this be, as current is directly proportional to voltage?


a question about singularities in gravity and Physics in general

I had this doubt bugging my mind for a long time about singularities in Physics. I heard that R.Penrose and S.Hawking have proposed that there could be singularities at Blackholes and at the time of the origin of universe (at the time of Bigbang). The concept of singularity is essentially a mathematical one. (unless you call anything that you cannot explain physically as a singularity). My question is (which i want to get it resolved once for all so that i stop bugging people with it :-))....We know that the gravitational field has a singularity at points where there are point masses, so is the case with point charges and electric field. So what is it they have proved about existence of singularities. I assume that they most probably have started with a universe which doesn't have point masses. If they did'nt start with point masses, then what did they actually started with. I want to know what they have started with in mathematical terms and physical terms, once for all. I am open to read some references as well to resolve this doubt. **EDIT :** I am more puzzled as to why I am not getting an answer to the question I've asked which doesn't involve any analysis but a plain request for information. Did they (R.P and S.H) assume existence of point masses in their model of universe (or objects or whatever it is) or they have assumed smooth matter distributions ? This is a fairly straight forward request for information....I do not understand why I don't get the answer for the question I've asked...rather I get something else.


A man running on the treadmill

Imagine a man is running on a treadmill. His inertia with respect to floor will be zero because he is not moving with respect to floor. If both he and the tread mill suddenly stops he will not fall, according to Newton's first law. Imagine an ant on the conveyor belt. He doesn't worry about our reasoning. He just believes in 1st law, but can't explain it using the 1st law. (Note: the ant will measure two momentum of the man. same but opposite in direction) Now imagine we are a 2D being, living in a closed world like the surface of the Earth. Is Newton's first law not applicable here? Then what will be the definition of inertia in this world? I think I should mention my current level of understanding. I read general relativity only in popular books.


Deriving diffusion coefficients from velocity field?

If I know the velocity, $\mathbf{v}(\mathbf{r},t)$, everywhere, is it possible to determine the diffusion coefficient, $D(\mathbf{r},t)$ everywhere as well? Would it be possible from using Fick's Law $$j = -D\cdot\nabla L$$ Or can it be found from the mean-squared displacement with $$\frac{\partial \langle r^2\rangle }{\partial t} = 4D$$ ($6D$ in 3D) What would be the best way to go about this?


Does a high energy photon experience deceleration or direction-change when it impinge into water or something due to the change of refraction index?

Wave experiences refraction when it propagates into another medium which has different refraction index. Lights surely does experience refraction at the border of mediums which have different refraction index because of its wave- like property. But I am wonder if high energy photons, whose energy exceeds MeV or GeV, also experiences refraction.


How can a black hole emit X-rays?

Considering that a black hole's gravity prevents light from escaping, how can a black hole emit X-rays? Visible light and X-rays are both electromagnetic radiation, shouldn't the black hole's gravity prevent X-rays from escaping?


What happen when two galaxies collide?

As the title suggests, I know collision of galaxies happens. Questions are   1. Why do they collide (obviously because of gravity, but in what conditions)?   2. Aftermath of the collision?   3. If there is a super massive blackhole in the center of the galaxies, what happen to them?


Why do we need different ensembles in statistical mechanics?

Why do we study these different ensembles, microcanonical, canonical, grand canonical ensemble ? Are they used for studying different physical system or scenarios?(e.g. in some system you can only treat it as mirocanonical, and in other cases you can only apply canonical ensemble) Do they have the same result at thermodynamic limit?


Transformation rule of a partial derivative

We know the following transformation rule: $$ \partial'_b = \frac{\partial}{\partial x'^b} = \frac{\partial x^c}{\partial x'^b} \, \frac{\partial}{\partial x^c} = \frac{\partial x^c}{\partial x'^b} \, \partial_c. $$ Here $\partial_c$ is covariant tensor of rank $1$. But what is the corresponding transformation rule for $$ \partial^m :=g^{mn}\partial_n?$$ How can we derive the transformation rule?


References on wave solutions in continuum mechanics

I am interested in literature on known wave solutions in continnum mechanics, precisely the following mechanical equation: $$\rho\partial_t^2u_i = C_{ijkl}\nabla_j\nabla_ku_{l}$$ My interest is spread around its study for cubic lattices, which doesn't seem straightforward to solve analytically to me. But I couldn't find a reference book or articles illustrating a peculiar solution of it.


Derivation of neutrino oscillation phase factor

As we know, the neutrino $\nu_{\alpha}$ with flavor $\alpha=e,\mu,\tau$ is a linear combination of mass eigenstates: $$ |\nu_{\alpha}\rangle=\sum_iU_{\alpha i}|\nu_i\rangle,\quad i=1,2,3 $$ where the propagation of the mass eigenstates $|\nu_i\rangle$ can be described by plane wave solutions: $$ |\nu_{i}(t)\rangle = e^{ -i ( E_{i} t - \vec{p}_{i} \cdot \vec{x}) } |\nu_{i}(0)\rangle. $$ Suppose that $t=T$ and $|\vec{x}|=L$. Then the phase factor can be given by $$ \begin{split} E_iT-p_iL &= E_i\frac{L}{\sqrt{1-m_i^2/E_i^2}}-\sqrt{E_i^2-m_i^2}L \\\ &=E_iL\left(1+\frac{m_i^2}{2E_i^2}\right)-E_iL\left(1-\frac{m_i^2}{2E_i^2}\right) +\mathcal{O}\left(\frac{m_i^4}{E_i^3}\right) \\\ &=\frac{m_i^2L}{E_i}+\mathcal{O}\left(\frac{m_i^4}{E_i^3}\right) \end{split} $$ However, this result is not correct. The artilce Neutrino oscillation on Wikipedia says that $$ |\nu_{i}(L)\rangle = e^{ -i m_{i}^2 L/2E }|\nu_{i}(0)\rangle $$ which differs in a factor of $1/2$. I can't figure out what is wrong with my derivation above (Maybe the formula for time interval $T$ is not correct, but I am not sure). Any help is greatly appreciated.


Information carried by single photon

In Quantum Information we can use photons for quantum bits (qubits). What i often read is that each photon can carry one unit of information, i.e. using the polarization state of a single photon.   I have two questions:   1) i read on this article that it is possible to send 1.63 bits of information per photon, what does that mean?   http://www.newscientist.com/article/dn13522-twisting-light-packs-more- information-into-one-photon.html#.Uy1cKBx22QE   2)If i can take a single photon state as a tensor product of its polarization state and its orbital angular momentum state (we could also add a frequency state), can i say that i am sending 2 (or 3) qubits of information in one single photon?   Thank you


Equivalent Chern Simons Theories

This is a follow-up question to FQH Edge Theory as decoupled chiral bosons . The document that I will be refering to is http://dao.mit.edu/~wen/pub/toprev.pdf . On page 14 in Eq.(2.33) the author argues that two Chern-Simons theories describe the same "Fractional quantum hall physics" when the K-matrices are the same up to conjugation with a SL(n,Z)-matrix. Also the charge vector of the two theories has to transform in a specific way. At the bottom of page 31 the author outlines a way to simultaneously diagonalizes the K-Matrix and the V-Matrix in the edge theory. Here however there doesn't seem to be a restriction that the conjugation matrices need to be in SL(n,Z). So I am wondering whether one may "accidentaly" perform a transformation on the fields that may leed to a theory describing a different fractional quantum hall state? For example we choose $K=\left( \begin{matrix} 1 & 2 \\\ 2 & 1 \end{matrix} \right)$ to describe a $\nu=2/3$ state. We diagonalize the matrix by conjugating it first with $\Lambda_{1}=\frac{1}{\sqrt{2}}\left( \begin{matrix} 1 & 1 \\\ 1 & -1 \end{matrix} \right)$ and then with $\Lambda_{2}=\left( \begin{matrix} \frac{1}{\sqrt{3}} & 0 \\\ 0 & 1 \end{matrix} \right)$. This yields a new K-Matrix K'=diag(1,-1). Then I am performing a Lorentz transformation that leaves K invariant and diagonalizes V. However none of these matrices is in SL(2,Z). (Also generically $U^{T}=U^{-1}$ is false in SL(2,Z).) I would be very happy if someone could clear up this confusion for me:) I am looking forward to your responses.


Why is the water on one side of a bridge choppy?

My daily commute take me across the Evergreen Point Floating Bridge on Lake Washington. I have noticed on several occasions that the water on the southern side of the bridge will be quite choppy while the water on the northern side is relatively still (not completely still but there is a noticeable difference from the other side). What causes this to happen?


What happens to objects pulled in by black hole?

What happens to the particles/elements/objects that sink into the gravitational pull and ultimately go to the interior part of a black hole? If, according to popular notion, it is crushed by the excessive gravitational high density thrust, what is that object reduced to? Which theory provides a basis for calculation of this effect?


How is Meissner effect explained by BCS theory?

Someone says we can derive the GL equations from BCS theory, which can explain Meissner effect, but I want a more clear physical picture of this phenomena.


Moon's orbit period as seen from a spaceship traveling at 0.8c

I am studying special relativity and I am trying to figure out the following small problem which occurred to me: An observer, the pilot of a spaceship flying to or from earth at v = 0.8c, is observing earth from his spaceship. He observes the moon orbiting the earth once every 45 days, 1.667 times slower than the period seen from earth. He also observes the earth is 1.667 times more massive than it seems to a person on earth. However, doesn't that mean the moon should orbit earth faster than normal, not slower? What is the solution to this problem?


Fields and Newton's Third Law

I'm studying basic physics. I'm using the text available at http://www.anselm.edu/internet/physics/cbphysics/downloadsI.html. It develops the universal law of gravitation by postulating the existence of a vector at each point of the form $$ g_P = \sum G\frac{m_i}{|r|_i^3}r_i, $$ Where $m_i$ and $r_i$ are the mass of and separation vector from $P$ for all particles that aren't at the point $P$. It examines the effect of one particle on another. If the separation vector is $r$ then from the above equation, we see that $$ g = G\frac{m_1}{|r|^3}r $$ and that when a particle of mass $m_2$ is placed at the given point, the force will be $$ F = G\frac{m_1m_2}{|r|^3}r $$ The authors then go on to claim that we can either repeat the development to see the effect of the second particle on the first one or apply Newton's third law. How does Newton's third law apply through a field? If it's the field exerting the force, then Newton's third law would require a force on the field and not the object "generating" the field, correct?


What does k denote in this equation?

Recently, I was attempting to compare the rate of flow of different liquids using the formulas given below: $$P = \rho g h$$ $$ \frac{dh}{dt}\varpropto h$$ $$ \frac{dh}{dt} = -k h $$ $$ \frac{dh}{h} = -k dt $$ $$ \int \frac{dh}{h} = -k \int dt$$ $$\ln h = -k t + c$$ My question here is: What does $k$ denote? Is it a physical quantity like density?


Finding electric field between overlapping surfaces

The problem is: > A sphere with radius R is centered at the origin, an infinite cylinder with > radius R has its axis along the z axis, and an infinite slab with thickness > 2R lies between the planes z=−R and z=R. The uniform volume densities of > these objects are ρ1, ρ2, and ρ3, respectively. The objects are superposed > on top of each other; the densities add where the objects overlap. > > How should the three densities be related so that the electric field is zero > everywhere throughout the volume of the sphere? What do you think the ratios > $\frac{\rho_{1}}{\rho_{2}}$, $\frac{\rho_{1}}{\rho_{2}}$ and > $\frac{\rho_{1}}{\rho_{2}}$ are equal to? > > Hint: Find a vector expression for the field inside each object, and then > use superposition. ![enter image description here](http://i.stack.imgur.com/wmKpo.jpg) The way I approached the problem: Find individual Electric fields and add them up to 0. Find the relation between the charge densities. To do that, I used Gauss' law. I bounded the region (the sphere) and that gives me $E_{sphere} = \frac{\rho_{1}R}{3\epsilon_{o}}$. However, because my region of interest is the sphere, I used the same bounding surface for the cylinder and the slab as well thereby getting the equations $E_{cylinder} = \frac{\rho_{2}R}{3\epsilon_{o}}$ and $E_{slab} = \frac{\rho_{3}R}{3\epsilon_{o}}$ Then, since the $E_{net} = 0$, $$E_{sphere} + E_{cylinder} + E_{slab} = 0$$ which, when simplified, gives $$\rho_{1} + \rho_{2} + \rho_{3} = 0$$ However, I'm not sure if a) what I've done so far is correct, b) how to proceed from here to calculate the ratios of the charge densities. Any help would be really appreciated. Thank you.


Falling through the rotating Earth

Suppose you were standing on the rotating Earth (not necessarily Equator or the poles) and suddenly your body lost the ability to avoid effortlessly passing through solid rock. Because the earth's rotation at the surface is considerably below escape velocity, you would slip below the earth's surface. If the earth's gravity were a consequence of a central point mass, you'd have an elliptical orbit (mostly) within the earth. With a planet of constant density, the gravity you feel underground is equivalent to standing on the surface of an identically dense planet with a radius equal to your current distance from the centre. So effectively, as you fall the gravity you experience lessens.   1. What would be the shape of the trajectory?    2. How close would you get to the centre?    3. How long would it take before your orbit brought you back to the surface (assuming no losses and a stationary planet)?   4. What complications would arise from the planet being in orbit around a star?   5. Bonus points for finding two well-known places on the Earth that you could travel between in one "orbit" using this method.


2d simulation shooting projectiles that inherit the gun's veocity at a moving target while also moving

Overview: I'm working on programming a simulation that requires 'shooting' projectile-type objects at other moving objects. How can I calculate the angle at which to shoot the object to hit? Details: Imagine you're holding some gun-type object that shoots projectiles at velocity $v_d$ in straight lines. You are moving with velocity $v_p$ at angle $\theta_p$, in a straight line. Another object, which we will name the target, is located a distance $R$ away from you at an angle $\theta_{tp}$, and is moving in a straight ine with velocity $v_t$ at angle $\theta_t$. I need to calculate the angle $\theta_a$ I should aim the gun in order to hit the target. Note that there is no decision on the TIME to fire. I have to fire immediately, at $t=0$, so this adds a helpful constraint but also makes it so that some targets are not able to be hit based on the numerical values of the parameters mentioned, which is fine. The target is moving so you need to lead it with your shot to compensate. Also, the projectile will inherit your velocity on top of its own since you are also moving, which will further modify the angle you need to shoot at. It seems to me that the strategy should be that I find a point in space which will be occupied by both the projectile and the target. I know all the points in space that the target will take along its linear path, and the times at which it will take all points. However, because the total velocity of the projectile varies based on the angle you shoot it (e.g. the magnitude of the velocity vector will be different depending on what Theta_a you decide to shoot at), I'm not sure how to figure out where to shoot it. Any help would be appreciated! In addition, if it is easy, a solution to a target moving in a circular path with radius $R$ (same $R$ as the distance between you and it, so it's making a perfect circle around you) would be appreciated!


direct conversion of heat to electric potential / current - are there theories that contradict the 'Kelvin Statement'?

In particular, William Thomson (Kelvin) appeared to be wrong about key things in physics (initially X-rays, aether, even aviation feasibility).   en.wikipedia.org/wiki/William_Thomson,_1st_Baron_Kelvin#Pronouncements_later_proven_to_be_false Are there any theories that contradict the idea that a heat differential must exist in order to convert heat energy to another form (causative) vs a heat differential being a secondary effect? I have seen some discussion about the potential for room-temperature thermionics (and whether or not that by definition relies on a heat differential to function?) jap.aip.org/resource/1/japiau/v94/i7/p4690_s1 If such a device existed to convert heat without a 'classical' heat differential being a primary causative of the conversion, what implications would it have for irreversibility? How does a purely homogeneous heat / kinetic energy pool in the classical sense behave when considering different atoms / elements with the same kinetic energy at the atomic level? Is it possible that different elements can never truly have the same kinetic energy in such a case, causing a situation where an exploitation can be found to do the conversion of energy? One hypothesis could be that heterogeneous atomic mixtures is the true cause of brownian motion (unless this has been proven to not be the case). In other words, is it not the case that homogeneous bulk matters behave predictably at any given kinetic energy level (heat)? This would have implications for the brownian ratchet being based on randomness, when in fact physical matter could potentially be biased to a predictable movement (even if only in a very confined system, i.e. nano-scale), where possibly extremely small kinetic differences can be engineered and exploited to generate electricity (or mechanical movement useful for work) from a single reservoir of heat, disproving the Kelvin statement.


What is the probability density function over time for a 1-D random walk on a line with boundaries?

If a single particle sits on an infinite line and undergoes a 1-D random walk, the probability density of its spatio-temporal evolution is captured by a 1-D gaussian distribution. \begin{align} P(x,t)&=\frac{1}{\sqrt{4 \pi D t}}e^{-\frac{(x-x_0)^2}{4Dt}} \end{align} However, suppose there are impassable boundaries on the line; on one side, or on both sides. Are there any boundary conditions for which there exists a closed form probability density function for how this particle will behave over time? Any references to such solutions would be extraordinarily helpful. EDIT. Attempting to generalize Emilio's result below for an arbitrary initial particle position $-L/2 < x_0 < L/2$. I had to work it out by example. I found the following "images" were required to account for reflections of an off-center particle at position $x_0$: for the first and second reflections on both sides the new gaussians had to be centered on ($-2L+x_0$, $-L-x_0$, $x_0$, $L-x_0$, $2L+x_0$). From the pattern I think the full solution can be expressed, for all integers $n$, as: \begin{align} P(x,t)&=\frac{1}{\sqrt{4 \pi D t}}\sum_{n=-\infty}^\infty e^{-\frac{(x-nL-(-1)^nx_0)^2}{4Dt}} \end{align} where the old $x_0$ is now defined as $nL+(-1)^{n}x_0$


Why is radioactive decay dependent on amount of substance available?

Radioactive decay is an attribute of unstable nucleus. When we represent it in equation, we don't involve any macroscopic attribute of substance. But still, rate of radioactive decay is proportional to amount of substance available (a macroscopic attribute). Why? What's really the relation with others when it comes to bleeding out its unstability?


Is it possible to float, starting at an elevated point, with a board and enough strength?

Premise: I have a structure like this.                  O      _____     |     |     |     |      I'm the `O`, and I'm sitting on a flat surface or board while being supported by two poles. (it looks like I'm floating, but just imagine I'm sitting. it's hard to draw stuff with only characters! :P) Now let's say one of the poles is removed.                  O      _____           |           |      Of course, if nothing else happened then the structure would fall over. But what if I pulled up on the board?                            O/ < left arm     right arm > /______                 \/ ^for|ce                        |      It's not a very good drawing, but my right arm is below the board, pulling up on it. Assuming that I was limitlessly strong, is it possible to not fall over? If so, is this possible?                             O     right arm > /___\ < left arm                 \/ \/      In other words, **is it possible to float, assuming you already start out elevated, with enough strength?** If not, why? My research has been quite useless, although I think I have found the name for this kind of thing: Bootstrapping. That article is not particularly helpful though. This question looks related, although I want to know whether this is theoretically possible with the aid of a board and when you are already elevated.


Euclidean Geometry in Classical Thought - Realization or Representation?

First post! :] This has been bothering me for a while now: [Taken from John J. Roche's "The Mathematics of Measurement: A Critical History"] > When physico-mathematicians in the seventeenth century needed to represent > their physical quantities mathematically they almost always turned to > geometry rather than to numbers. The author goes on to explain this further, saying how the Ancient Greeks had used geometrical figures to represent physical quantities rather than numbers (influencing the physicists later on) and how they were most appropriate for representation since they denoted "continuous physical quantities". I don't understand this; what makes physical quantities continuous? Wouldn't you figure them to be discrete considering that they are quantified by units? More importantly, is the author saying that physicists and mathematicians realized physical relationships geometrically? Or is he just saying that they merely chose to _represent_ physical quantities and their relationships to others with geometrical figures? It seems extremely non-intuitive to view physical quantities as being segments/planes/solids instead of as quantifiable values with defined dimensions. Of course it's easier to view/realize proportionalities between physical quantities through geometrical/graphical means, but there must have been a numerically-based intuition even _before_ this. In other words in finding a relationship between 2 quantities, there must have been a numerically-based line of thought that preceded a geometrical _demonstration_ of it, simply through my reasoning that it is more intuitive. I wanted to discuss this though, to see if anyone had any input/corrections to make about my thoughts here; I'd love to hear what you guys have to say :] Here's a direct link to the book I'm referring to (Google Books). The quote stated above is on page 87, and it continues on from there. Thank you!


Angular momentum of two rotating spheres

I am trying to calculate an instantaneous merger of two rotating spheres into one. Two spheres each rotating around their own axis of rotation (which are generally not aligned) and moving relative to each other. Imagine two entirely inelastically colliding planets. While I can calculate their individual spin angular momentum and the orbital angular momentum, it is unclear to me how to combine this into one (spin) angular momentum for the resulting sphere. It seems to me I cannot simply add the four vectors together... or can I? After merging the two into a single body maintaining the initial momentum, can I calculate the released energy (in the inelastic collision) simply as the difference in kinetic energy, using the same reference frame? I realize this should be pretty trivial, but I have thus far not been able to neither derive nor locate and answer.


Question regarding 2 conducting spheres connected by a long fine line (electric potentional)

I'm having a bit of trouble understanding a pretty simple issue. Assuming I have 2 conducting spheres uniformly charged connected by a long fine line (as shows in the added photo) and im being asked about what will happened, how much charged will move from on spheres to the other. I was told that the electric potential of these 2 spheres are: $k\frac{Q_1}{R_1}$ and $k\frac{Q_1}{R_1}$ The thing is, that I don't get why, I mean, $R_1$ and $R_2$ are the radiuses of the 2 spheres, these are not vectors or anything and they do not meet the same coordinates and this is what I should take when calculating the potential using integral the good way... So, how can $k\frac{Q}{R}$ be the potential of the spheres? ![enter image description here](http://i.stack.imgur.com/aDzN4.gif)


Physics Paradox about Newtons Second Law $F=ma$

I randomly thought of this- I'm surprised it took me so long to think of it, and I'm SURE I'm making a simple oversight of something completely trivial, but here it goes. (Im only in high school taking normal physics, so don't laugh if this is stupid.)   1. If force equals mass times acceleration, wouldn't a basketball dropped from the top of the Eiffel tower exert the same force on the ground as a basketball dropped a foot off the ground? They both have the same mass, and they both are accelerating towards the ground at a rate of g=9.81 m/s^2. (I don't know what terminal velocity is that well as im only in physics 1 in highschool, but just assume that air drag is not important and the ball doesn't reach terminal velocity.)   2. Also, if a ball IS dropped high enough to reach terminal velocity, then it accelerates at 0 m/s^2, so it has a force of ZERO when it hits the ground?


what is the relationship between these two sorts of anomalies?

Recently there has been a few questions about anomalies in QFTs: Why do some anomalies (only) lead to inconsistent quantum field theories Classical and quantum anomalies In these, people have been discussing anomalies that (let me know if I am wrong in my understanding here) come from essential symmetries which cannot be realised by finite energy unitary representations on a suitable Hilbert space. Meanwhile, my understanding of "anomaly cancellation formulas" that appear for instance in 11d supergravity is that, a priori the action functional is not an honest function on the configuration space of the theory, but a section of a line bundle over the config space. Nontriviality of the line bundle prevents integration of the action functional. This is my definition of an anomaly. Anomaly cancellation happens for example when the bosonic and fermionic sectors have action functionals in dual line bundles, so that in the tensor product theory there is no anomaly, though either sector is by itself anomalous. My question is : what is the relationship (if any) between these two perspectives? I suspect there is some on the basis of the Borel-Weil-Bott theorem, but this is idle speculation. Thanks.


Finding Surface Tension of water at certain Temperature and Pressure

So the question is: > Using the Young-Laplace Equation (if applicable), find the surface tension > (dynes/cm) for water at 20 degrees Celsius with 2.5 psi. Round to the > nearest tenth. Well, I didn't use the Young-Laplace equation, not sure if needed though. What I did was use the Eötvös rule and it's special case for water to solve the question. The equation is: $$\gamma = 0.07275\;\frac{N}{m}\;\times\;(1-0.002\times(T-291K))$$ What I did was convert 20 Celsius to Kelvin (293K) and then put in the equation to get: $$\gamma = 0.07275\;\frac{N}{m}\;\times\;(1-0.002\times(293K-291K))= 0.072459\frac{N}{m}$$ However, I think I may be wrong as this does not account for pressure at all. Which ends up becoming about $72.46\frac{dynes}{cm}$ Am I right or wrong? And is there a better/correct way of doing this?


Is there a "present state" of distant stars if simultaneity is relative?

Special relativity theory says simultaneity is relative, meaning that different observers will not agree on what happened first and what second. Does it then make sense to say that looking at distant stars, we see them how they looked "billions of years ago" and not how they look now? Does it make sense to talk about what these stars look like _now_? How do we define this "now" if simultaneity is relative?


How do 2 magnets spin by themselves if positioned precisely?

A few years ago I went to a museum, where there was a board with 2 bar magnets, on a pole each so they could rotate. If you rotated them so the lined up with the same poles (N) facing each other, they'd repel spinning until the south poles lined up, then they'd repel and spin so the north poles lined up and so. This was without any extra help, once you lined them up they would just keep spinning by their selves. How exactly is this possible that the 2 magnets kept producing kinetic energy?


Are higher order mixed partial derivatives of wave function with different ordination equal?

For example, given two operators:   $$A = \frac{\partial}{\partial x}+\frac{\partial}{\partial y},$$   $$B =\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2} + 1.$$ Deriving commutator $[A,B]=0$ involves mixed partial derivative of higher order with opposite signs, and it is very temptative to zero them to make commutator equals zero. But is this allowed, i.e., does any valid quantum wave function must have continues (mixed) partial derivatives of higher order so one can use Schwartz theorem?


Coulomb force on the center of a hexagon?

Six point charges $q$ are at the corners of a regular hexagon that has sides of length $a$. What is the force on another charge $Q$ which is located in the center of the hexagon? What is the force on $Q$ if we remove one charge $q$ from one corner? * * * I know I'm supposed to use Coulombs law and that the force on $Q$ is the sum of all other forces from the six point charges $q$. But how do I compute the forces on each of the six points? Thanks much.


Is there valid physics behind the bodybugg?

The bodybugg is a faddish gadget whose marketers claim it can measure your body's daily energy expenditure. Their sales literature says: > As reported in the British Journal of Sports Medicine, 2008 edition, the > bodybugg® armband achieved average accuracy rates of 90% in studies > comparing calorie expenditure measurements from the bodybugg device to > calorie expenditure measurements in a metabolic chamber (the "gold standard" > of calorie expenditure measurement). "Accuracy rates of 90%" is obviously not how anyone with any level of science literacy would describe error bars. Their FAQ says: > In addition to measuring motion and steps, the bodybugg is able to see how > much you are sweating, your skin temperature and the rate at which heat is > being dissipated from your body. They also say it doesn't measure your heart rate. Is there valid physics behind this device, and if so, what principles would it be based on? Measuring "motion and steps" would seem to involve an accelerometer, but it seems unlikely that that would be able to tell the difference between, say, walking on a horizontal sidewalk and walking uphill. How can they separately measure skin temp and rate of heat dissipation? Maybe there's a thermometer in contact with your skin and one sensing the air temperature? It's not obvious to me how one could combine the mechanical data with the thermodynamic data.


How can a Higgs decay to heavier products than its mass?

How is it possible that a higgs at ~125 GeV can decay into 2 W bosons @ ~ 80 GeV a piece (for example)? Shouldn't a particle only be allowed to decay to lighter particles + energy? Diagram copied from this question ![enter link description here](http://i.stack.imgur.com/eljaD.jpg)


Can the two electromagnetic field tensors be combined into a more general tensor?

Given the electromagnetic field tensor $$\begin{align} F_{\mu\nu} = \begin{pmatrix} 0 & -E_{x} & -E_{y} & -E_{z} \\\ E_{x} & 0 & B_{z} & -B_{y} \\\ E_{y} & -B_{z} & 0 & B_{x} \\\ E_{z} & B_{y} & -B_{x} & 0 \end{pmatrix} \end{align}$$ and its dual $$\begin{align} G_{\mu\nu} = \begin{pmatrix} 0 & B_{x} & B_{y} & B_{z} \\\ -B_{x} & 0 & E_{z} & -E_{y} \\\ -B_{y} & -E_{z} & 0 & E_{x} \\\ -B_{z} & E_{y} & -E_{x} & 0 \end{pmatrix} \,, \end{align}$$ does there exist a general tensor $H_{\mu\nu}$ that combines $F_{\mu\nu}$ and $G_{\mu\nu}$ into a more general tensor in Minikowski space?


Quantum Mechanical States

What can be the precise answer to the question that > Quantum states are complex and infinite dimensional. Why is this so? Is it because they belong to the complex Hilbert space? Even if they belong to the Hilbert space, why they are infinite dimensional, is this because of the projection operator?


What should I look for at SAR levels for cell phones?

I've been trying to wrap my head around SAR levels when comparing cell phones. Considering that I spent most of my time not talking on the phone but carrying it around near my body, would a phone **lower body** value but **higher head** value be preferable one with the reverse combination? Or, suppose one phone has a value that is 2x of another. How should I think of it? Is it significantly worse or slightly worse?


Work done against gravity

The work done against gravity is $mgh$, well at least that's what my textbook says. I have a question: I can apply a force say 50N, so total work done = $mgh + mah$. Where $ma$ = Force. But the truth is irrespective of the force applied, the work done against gravity is always $mgh$. Why? For example, when I move an object with a force, the work done is more, so work depends on the Force. But in case of gravity it is always depends upon the weight


Do photons interact with each other, or with themselves only?

Even if the double slit experiment gives interesting (weird) results, it only concludes that each photon interacts with itself after passing the two slits. I have been thinking about a different experimental setup, where you have two well defined light sources (with specific wave lengths and phase) but no slits. And now to my questions: Has anyone ever done such an experiment, and will there be an interference pattern on the wall? If the answer to the second question is "no", light can not be a true wave - it only has some wavelike properties. But if it is "yes", things become much more interesting. If there is an interference pattern on the wall, there has to be an interference pattern even if both light sources are emitting single photons at random, but as seldom as, say, once per minute. That in turn would mean that the photons know about each other, even if they are separated in time with several seconds, and the light sources are independent (not entangled).


Thomson's jumping ring experiment

For the jumping ring experiment (I think it is also called Thomson ring experiment) why does the ring float and not move up and down the ring as would be expected, I think by Lenz's law, since would the ring when the current thru the coil is decreased (due to it been ac) not fall back down to increase the magnetic flux thru it that has been lowered due to the smaller current??


Why does Snell's law not factor in wavelength?

If dispersion is caused due to different wavelengths bending with different angles, how is Snell's law right in generalizing it as $\sin(i) / \sin(r)$? Or am I missing something?


The Deflecting System in a Hot Cathode Ray Tube

In an HCR-Tube, the deflecting system used to deviate the _electron_ beam is made of _positively_ charged plates. How is this justified? If, due to some malfunction, the electron beam deflects from its original path before it is actually supposed to, won't it stick to the plates? And why is it not wiser to use _negatively_ charged plates? If this is done, then the electrons of the beam will return to their stream even if they are accidentally deflected, and when the actual deflection is required positive plates can be introduced.   So why only positive plates?


Dealing with dirac notation with regards to different basis'

So this should be a pretty simple question. So we say that $\langle x | \psi \rangle = \psi(x)$. In other words $\psi(x)$ is the ket $|\psi\rangle$ expressed in terms of the $x$ basis. Now suppose I wanted to get $\psi(x+a)$. How would I act on $|\psi\rangle$ to get it? I think I could just do $$ \langle x' | \psi \rangle = \psi(x+a)$$ Where $x' = x+a$. But I don't really know what this means in terms of how $\langle x' |$ and $\langle x |$ are related, or the formal justification for this. Perhaps my trouble is I don't thoroughly understand what the $x$-basis is or what $\langle x | \psi \rangle$ is really doing.


Lorentz transformations for sine wave in random direction

Hello i have been struggling to derive Lorentz transformations for a sine wave, which is traveling at random direction. I started by prooving that phase $\phi$ is invariant for relativity and that equation $\phi = \phi'$ holds. By using the above equation i am now trying to derive Lorentz transformations for angular frequency $\omega$, and all three components of the wave vector $k$, which are $k_x$, $k_y$ and $k_z$. This is my attempt: \begin{align}\phi' &= \phi\\\ \omega'\Delta t' + k' \Delta r' &= \omega \Delta t + k \Delta r\\\ \omega'\Delta t' + [{k_x}' , {k_y}' , {k_z}'] [\Delta x' , \Delta y' , \Delta z'] &= \omega \Delta t + [k_x , k_y , k_z][\Delta x , \Delta y , \Delta z]\\\ \omega' \Delta t' + {k_x}'\Delta x' + {k_y}' \Delta y' + {k_z}' \Delta z' &= \omega \Delta t + k_x \Delta x + k_y \Delta y + k_z \Delta z \end{align} Now with just the left side: \begin{gather} \omega' \gamma \left(\Delta t - \Delta x \frac{u}{c^2}\right) + {k_x}' \gamma \Bigl(\Delta x - u\Delta t \Bigl) + {k_y}' \Delta y + {k_z}' \Delta z\\\ \gamma \left(\omega' \Delta t - \omega' \Delta x \frac{u}{c^2}\right) + \gamma\Bigl({k_x}' \Delta x - {k_x}' u \Delta t \Bigl) + {k_y}' \Delta y + {k_z}' \Delta z\\\ \gamma \left(\omega' \Delta t - {k_x}'c\, \, c \Delta t \, \frac{u}{c^ 2}\right) + \gamma \Bigl({k_x}' \Delta x - \frac{\omega'}{c} u\frac{\Delta x}{c} \Bigl) + {k_y}' \Delta y + {k_z}' \Delta z\\\ \Delta t \, \gamma \Bigl(\omega' - {k_x}' u \Bigl) + \Delta x \, \gamma \Bigl({k_x}' - \omega' \frac{u}{c^2} \Bigl) + {k_y}' \Delta y + {k_z}' \Delta z \end{gather} From this I can write down the Lorentz transformations. \begin{equation} \begin{split} \gamma\Bigl(\omega' - {k_x}' u \Bigl) &= \omega\\\ \gamma \Bigl({k_x}' - \omega' \frac{u}{c^2} \Bigl) &= k_x\\\ {k_y}' &= k_y\\\ {k_z}' &= k_z\\\ \end{split} \end{equation} My professor said that my signs are wrong, but what am i doing wrong?


Entanglement entropy and area law

I am currently reading a review "Area law for the entanglement entropy" by Eisert, Cramer and Plenio (2010). From what I understand:   1. In one dimension, for local gapped models, we have an area law for entanglement entropy.   2. In one dimension, some models with long range interactions + all critical systems obey a log(N) law, where N is the size of the subregion of the 1D space, the pre-factor depends for CFT models on the central charge.   3. For higher dimensions, local gapped (equivalently quasi-free) models obey an area law.   4. For higher dimensions, critical bosonic models obey an area law, fermionic ones obey a divergent log law with pre-factor dependent on the topology of the Fermi surface (which is also probably related to the central charge of the CFT, but no exact results are known). However I am a bit confused about the treatement of the Klein-Gordon field. The massive Klein-Gordon field is a local gapped model, hence not critical. Papers by Bombelli and Srednicki show that entanglement entropy obeys an area law for dimension greater than 2, in agreement with point 3. However in the review they seem to say that this is a critical system. Thanks to point 4 there is still no contradiction with the area law result. However, this becomes problematic for the 1D case: a massive or massless KG system obeys a divergent log law, as shown here: http://arxiv.org/pdf/hep- th/9401072.pdf and in the review. To me this is only true if the field is massless. A massive KG field is a local gapped system, so not critical and hence it should follow an area law according to point 1, not a divergent log law. So it seems to me that the problem is that these papers consider the KG field (massive or massless) as critical, whereas to me only a massless KG field is critical. What did I misunderstand?


What is the geometry of our universe?

I have some questions about the notion of space in current cosmological theories. I have been trying to decipher a few papers written on this topic such as this one, but I keep running into some trouble. Most of the papers I have been skimming talk about anti-de Sitter space (AdS) and about how the best definition of nothing is > _the limit of anti-de Sitter space in which the curvature length goes to > zero._ This is great and all, but I have read on Wikipedia that AdS implies a negative cosmological constant, which would imply that dark energy should be attractive. Is my reasoning here incorrect? If not, could someone explain why cosmologists spend so much time devoted to studying AdS when the universe is clearly expanding? I have read somewhere about local geometry vs. global geometry, but I wasn't sure if that is what is going on here. The papers are so complex.


Divergence theorem in complex coordinates

This question is related to Stokes' theorem in complex coordinates (CFT) but, I still don't understand :( Namely how to prove the divergence theorem in complex coordinate in Eq (2.1.9) in Polchinski's string theory $$\int_R d^2 z (\partial_z v^z + \partial_{\bar{z}} v^{\bar{z}})= i \oint_{\partial R} (v^z d \bar{z} - v^{\bar{z}} dz ) (1) $$ I may try $$ \int_R dx dy ( \partial_x F_y - \partial_y F_x) = \oint_{\partial R} (F_x dx + F_y dy)(2) $$, but what kind of substitution I should use to get Eq. (1)?


Does the potential energy for a given photon increase or decrease in quanta?

As a photon leaves a strong gravitational field, it loses energy and redshifts. Is the exchange in potential energy of a photon characterized by energy quanta?


Existence of negative temperatures and the definition of entropy

How negative temperatures can be possible has been treated on StackExchange before (several times in fact), but in light of some recent academic discussion, most of these answers seem to be possibly wrong or incomplete. The literature I am referring to is Dunkel & Hilbert, _Nature Physics_ **10** , 67 (2014) arXiv.1304.2066, where as I understand it, it is shown that negative temperatures are an artefact of choosing an incorrect definition of entropy. The Wikipedia article on the matter has also been amended to reflect this. This was later challenged by similarly well-known scientists, in arXiv:1403.4299 where it was, among other things, pointed out that this argument is actually decades old (citing Berdichevsky _et al._ , _Phys. Rev. A_ **43** , 2050 (1991)). The original authors quickly countered the arguments made in the comment, by what seems to be a rigorous treatment of the matter, arXiv:1403.6058. The first arXiv comment (arXiv:1403.4299) has been updated since and it still reads that "Obviously severe points of disagreement remain". What I am asking, then, is whether someone on StackExchange might be able to shed some light on the matter as to how there can be a disagreement about something that seems should be a mathematical fact. I would also be interested in hearing whether changing the definition of entropy from that of Boltzmann to that due to Gibbs might potentially change any other results. Might for example the Wang-Landau algorithm be affected seeing that it does use the density of states and that you can never simulate infinite systems (although as I understand it, even in the present context with finite scaling you should be able to get consistent results)?


Intermolecular Hydrogen Bonding

I learnt about intermolecular hydrogen bonding today, which occurs between molecules such as ortho-nitro-phenol. What I was told is that in case of intermolecular bonding, the molecules separate from each other, opposite of what happens during intramolecular hydrogen bonding. I don't understand why is this. Why would attraction within a molecule cause separation of the molecules? There is obviously some intramolecular hydrogen bonding present in this case, but I understand that this might be negligible. Still, why should the molecules spread away from each other?


Does a four-divergence extra term in a Lagrangian density matter to the field equations?

Greiner in his book "Field Quantization" page 173, eq.(7.11) did this calculation: > ${\mathcal L}^\prime=-\frac{1}{2}\partial_\mu A_\nu\partial^\mu > A^\nu+\frac{1}{2}\partial_\mu A_\nu\partial^\nu > A^\mu-\frac{1}{2}\partial_\mu A^\mu\partial_\nu A^\nu $   >  $\space\space\space\space=-\frac{1}{2}\partial_\mu A_\nu\partial^\mu > A^\nu+\frac{1}{2}\partial_\mu[A_\nu(\partial^\nu A^\mu)-(\partial_\nu A^\nu) > A^\mu]$ > > The last term is a four-divergence which has no influence on the field > equations. Thus the dynamics of the electromagnetic field (in the Lorentz > gauge) can be described by the simple Lagrangian > > ${\mathcal L}^{\prime\prime}=-\frac{1}{2}\partial_\mu A_\nu\partial^\mu > A^\nu$ Yes, **if it is a four-divergence of a vector whose 0-component doesn't contain time derivatives of the field** , indeed according to the variational principle this four-divergence will not influence the field equation. And actually I calculated the time-derivative dependence of 0-component of $[A_\nu(\partial^\nu A^\mu)-(\partial_\nu A^\nu) A^\mu]$, in which only $[A_0(\partial^0 A^0)-(\partial_0 A^0) A^0]$ could possibly contain time- derivative, which vanishes fortunately, so whatever the general case it doesn't matter in this present case. But how can he seem to claim that **it holds for a general four-divergence term** ,`The last term is a four-divergence which has no influence on the field equations`? EDIT:   I only assumed the boundary condition to be $A^\mu=0$ at spatial infinity, not at time infinity. And the variation of the action $S = \int_{t_1}^{t_2}L \, dt$ is due to the variation of fields which vanish at time, $\delta A^\mu(\mathbf x,t_1)=\delta A^\mu(\mathbf x,t_2)=0$, not having the knowledge of $\delta \dot A^\mu(\mathbf x,t_1)$ and $\delta \dot A^\mu(\mathbf x,t_2)$, which don't vanish generally, so the four-divergence term will in general contribute to the action, $$\delta S_j=\delta \int_{t_1}^{t_2}dt\int d^3\mathbf x \partial_\mu j(A(x),\nabla A(x),\dot A(x))^\mu =\delta \int_{t_1}^{t_2}dt\int d^3\mathbf x \dot j^0 =\int d^3\mathbf x [\delta j(\mathbf x, t_2)^0-\delta j(\mathbf x, t_1)^0]$$ which does not vanish in general!


Is Schrodinger's Cat a real conceptual problem or just a problem with approximations?

In this thought-experiment a cat is placed in a box set with a bottle poison that will release and kill it depending on whether or not a certain radioactive particle decays. The box is kept closed and it is asked "is the cat alive or dead?" Since the decay of the particle is a quantum mechanical process- it is represented by a state function. This function remains in a "superposition state" until it is observed whether the particle has decayed or not. The cat is presumably and unfortunately entangled with this function- it's own life/death is in a superposition state until it is indeed observed to have dodged the bullet or not. Schrodinger used this thought experiment in an attempt to show the ridiculousness of certain aspects of quantum theory. His motives were clear in devising it and are well documented in his correspondences with Einstein. My question is not about the cat, the particle, or Schrodinger. I want to know about the box. The box that contains this scenario must effectively isolate the outside observer from any information about what is (or could be is perhaps more correct) going inside of it. So what sort of box would qualify? It seems that such a box could not have any interaction at all with the outside universe- no gravitational or thermal effects- anything. Otherwise would not some kind of information about what was going on inside escape without it being opened? And this being said- were such a box constructed would it not be entirely cut off from reality and how could we be sure it even contained a cat still or was even there at all? Is this experiment possible or are the problems I outline above a deterrent to it's physical possibility?


According to Srednicki's QFT textbook, page151 "According to our development of LSZ formula in section5, each incoming and outgoing particle should correspond to a single-particle state that is an exact eigenstate of the exact hamiltonian. This is clearly not the case for a particle that can decay." Why an unstable particle is not the exact eigenstate of the exact hamiltonian? Because its eigenvalue of the full hamiltonian is the $m_\phi$. Where is my fault?


Why does microwaved food get cold faster

My evidence is entirely anectodal and non-scientific, but I've noticed food gets cold faster when it's been heated in a microwave instead of a stove. Is this true? And if it is, why does it happen?


What is the electric field between and outside infinite parallel plates?

I know that Gauss's law says $$\oint_S {\vec{E} \cdot d\vec{A} = \frac{q_{enc}}{{\epsilon _0 }}}$$ and that because $\vec{E}$ is always parallel to $d\vec{A}$ in this case, and $\vec{E}$ is a constant, it can be rewritten as $$\left | \vec{E} \right |\oint_S {\left | d\vec{A} \right | = \frac{q_{enc}}{{\epsilon _0 }}}$$ which is also equal to the electric flux through a Gaussian surface. I've included a picture to make it easier to ask my question. ![Parallel plates of uniform, opposite charge densities with Gaussian cylinders I - IV](http://i.stack.imgur.com/Mloif.png) What I don't get is how, mathematically, there is no electric field "outside" of the plates and how the electric field between them is determined. I - IV are Gaussian cylinders with one face on a plate. $$\vec{E} = \vec{E_+} + \vec{E_-}$$ Where $\vec{E_+}$ is the electric field from the positive plate and $\vec{E_-}$ is the electric field from the negative plate. For I: $$\left | \vec{E_+} \right | \pi r^2 = \frac{\sigma \pi r^2}{{\epsilon _0 }}$$ $$\left | \vec{E_+} \right | = \frac{\sigma}{\epsilon_0}$$ $$\left | \vec{E_-} \right | \pi r^2 = \frac{0}{{\epsilon _0 }}$$ $$\left | \vec{E_-} \right | = 0$$ This doesn't make sense to me because it's saying that the magnitude of the electric field due to the negative plate is 0, but even if I just assume that's because Gauss's law only works for surfaces that enclose some charge and disregard the 0 I got for the negative electric field, I'm still confused for the following reason: For III: $$\left | \vec{E_+} \right | \pi r^2 = \frac{0}{{\epsilon _0 }}$$ $$\left | \vec{E_+} \right | = 0$$ $$\left | \vec{E_-} \right | \pi r^2 = \frac{-\sigma \pi r^2}{{\epsilon _0 }}$$ $$\left | \vec{E_-} \right | = \frac{-\sigma}{\epsilon_0}$$ Now I have values for $\left | \vec{E_+} \right |$ and $\left | \vec{E_-} \right |$, but when they're going in the same direction (as they are between the plates), they sum to 0, which isn't right. To the left, when you add them going in opposite directions, you get $\frac{2\sigma}{\epsilon_0}$ and to the right you get the same thing. What am I doing wrong?


How much of current flows through a bird sitting on a power line?

I've been googling for hours and went through over a hundred answers. Now, some say the bird doesn't form a closed loop, some say the current is so small that it doesn't kill the bird. From as much as I understand electricity, I'm sure the bird does make a parallel connection to the wire, so there must be some electrons moving through it's body. Could someone please explain which is the right answer and why? And if it's the case that a parallel circuit is formed, how can I calculate the current going through the birds body?


How does light get from one bulb to another in a sequence of light bulb?

How does light get from one bulb to another in a sequence of light? Like in the advertisements or a smaller group? Is it the same light?


Calculate the acceleration of a moving elevator

An elevator is moving down with uniform acceleration. A man inside the elevator who wants to measure the acceleration drops a coin as soon as the elevator starts. The coin is 6ft above the floor of the elevator at the moment it is dropped. It takes 1 second for the coin to reach the floor of the elevator. Calculate the acceleration from all this data. How do I go about solving it ? I have absolutely no idea. Can anyone give at least some hints ?


Is there any literature that compares the candle-power to the candela?

I am unable to find anything in the literature to back up the claim that appears all over the internet that:               The modern candlepower now equates directly to the number of candelas.      Can anyone please suggest something in the scientific literature to back up this statement? If this is not the correct stack exchange for this question, please suggest a better one.


Physics of the inverted bottle dispenser

![Look at the red arrow](http://i.stack.imgur.com/aBnic.jpg) When you invert a water-bottle in a container, the water rises and then stops at a particular level --- as soon as it touches the hole of the inverted bottle. This will happen no matter how long your water-bottle is. I understand this happens, because once the water level touches the hole, air from outside cant go inside and therefore there is nothing to displace the water that falls out of the container. Now according to the laws of pressure ---- the pressure at the water level must be same everywhere --- whether it's inside the water bottle or outside. And that must be equal to the atmospheric pressure. Therefore the pressure of the water column + air column inside the inverted bottle must be equal to the atmospheric pressure. What I dont understand is, no matter how long a bottle you take, the water level will always stop at the hole. So that means that no matter how long a bottle you take, the pressure of the water column + air column inside the water bottle will be equal to the atmospheric pressure. How could this be possible? Also I'd like to let you know that, if you pierce the upper part of the bottle with a small pin, then the water level rises and overflows out of the container. I'm assuming air from outside rushes in and pushes the water out.


NP-completeness of non-planar Ising model versus polynomial time eigenvalue algorithms

From the papers by Barahona and Istrail I understand that a combinatorial approach is followed to prove the NP-completeness of non-planar Ising models. Basic idea is non-planarity here. On the other hand, we have polynomial time algorithms for calculating eigenvalues for matrices. This confuses me. Should not be solving an Ising model be equivalent to calculating the eigenvalue of the Hamiltonian matrix of that model? In that case why is it NP-complete? Is coming up with such a Hamiltonian matrix at the first hand is NP-complete? If it is so, it makes the whole project (coming up with such Hamiltonian and solving it) NP-complete.


Paradox?: What is the form of radiation experienced by a harmonically accelerated observer?

Theory predicts that uniform acceleration leads to experiencing thermal radiation (so called Fulling Davies Unruh radiation), associated with the appearance of an event horizon. For non uniform but unidirectional acceleration the shape of the experienced radiation changes from thermal to other spectral densities, but also is predicted to exist. But suppose the acceleration is periodic and oscillatory, i.e. no permanent horizon persists? In particular, what about the case of harmonic motion, for a full cycle, half a cycle, etc.? Here is an even simpler related problem that makes the apparent paradox easier to see. Suppose at proper time t=0, I accelerate at constant acceleration k in the x direction for t0 seconds, presumably experiencing Unruh radiation. Then I accelerate with acceleration -k, (in the -x direction,) for 2*t0 seconds, seeing more Unruh radiation coming from the opposite direction, and then I finish with with acceleration +k for the final t0 seconds. At the end of the 4*t0 proper seconds, I'm back where I started, at rest, without any event horizon. Was the Unruh radiation I felt when reversing acceleration secretly correlated or entangled with the radiation I initially and finally saw? Otherwise, from a more macro scale, I didn't actually necessarily move much, and the acceleration event horizon was instantaneous, evanescent and fleeting, so whence arose the Unruh radiation?


Collision of multiple fluid flows at a point

I would like to do an analysis when two or more fluid streams flowing from different direction with different velocity, meet at a point. What happen after the collision? Are there any related studies or papers?


Is there a standard reference where entropy is set equal to zero in property tables?

For practical considerations, it seems that entropy is only meaningful as a difference between states, like $\Delta s$ going from state A to state B. For an ideal gas, for instance, standard formulas give some infinities for entropy at absolute zero. But we still tabulate values. Looking at these tables, I see the s=0 point to be around -10 to -5 degrees C for air. This doesn't seem to be consistent with other substances (notable steam tables). Is there a commonly accepted rule about where entropy to zero. While that point is arbitrary physically, different steam tables would need to use the same offset to be compatible among each other. I'm interested to know if there's a standard rule for this. Particularly for ideal gases.


Why does my gravity simulation do this?

For a school project i created a simple 2D gravity sim in Matlab using the simplest possible method. There are 2 nested loops so that the total force and acceleration of every object can be calculated. I found that conservation of momentum in x and y works perfectly. I also use a damping so if two object are very close, their acceleration doesn't explode. Then i place two objects of equal mass on coördinates (-100,0) and (100,0) with initial speeds 0. If i start the sim, the two objects will move towards eachother, pass though eachother and move away again. However, they move away much farther than where they originally started. Would that mean that kynetic energy + pot energy is violated, or is this normal? In the real world this obviously never happens because objects can't pass through eachother. It is not very important to solve the problem, the project is from a few months ago. I just wonder what went wrong. Here is my code. The circle.m must be in the same folder as the gravity.m Some names of variables are in Dutch, but i translated the comments. http://speedy.sh/4wVqm/Matlab-gravity2.rar


Relativistic Computation?

Is it possible to employ relativity to develop computational technology? Here is a really basic example: Build a Computer and Feed it the Problem (say the problem is projected to take 10 years to solve) Start the solution of the problem using algorithm. Now go on to your convenient relativistic jet... accelerate very near the speed of light (utilizing time dilation to your advantage) and make a round trip back to the planet, which takes a reasonable amount of time for the observer within the relativistic jet to complete (say a couple weeks) and allows them to arrive back to their computer with the problem solved. This sort of system could solve very difficult problems "relatively" quickly in the sense that the user would be able to find the solution to their problem in much less time ,relative to themselves, then normal. A more user convenient system can be found if research is being conducted in a heavy gravitational field (time speeds up). For example: If you have a lab close to the surface of a black hole, and you send your computer far away (much less gravitational attraction) a mere couple of minutes could pass in your lab's frame of reference while the computer had thousands of years to work, in it's frame of reference, so then the computer can be fetched back to the lab with the solution found. Unfortunately Earth's gravitational time dilation is not nearly enough to take advantage of this sort of system (which is convenient in that it allows the user to remain stationary). Are there any better ideas? Just for the sake of though experiment and logic.


What causes a soccer ball to follow a curved path?

Soccer players kick the ball in a linear kick, though you find it to turn sideways, not even in one direction. Just mid air it changes that curve's direction. Any physical explanation? Maybe this video will better emphasize this phenomenon: http://www.youtube.com/watch?feature=player_detailpage&v=ZEv7QEFNVq0#t=265s


Heating a black body

By definition, Black body has absotivity=emmisstivity=1. This means the black body radiates all energy it accepts. Does this mean the black body cant be heated?


The physics behind The Great Flood

The book of Genesis floats (pardon the pun) some interesting numbers when discussing the Great Flood. For example, it rained for 40 days and 40 nights, and at the end of that time, the entire planet was covered in water. I think we can deduce how much water that would have had to be, estimating that the highest peaks in the Himalayas were covered with water. (8,848 meters above sea level) My questions are, how fast would the rain have had to come to raise the ocean level that high in 40 days and nights, how much would the mass of the earth have changed for this event, and would that significantly alter the strength of gravity on earth?


Beam splitters and Mach-Zender interferometers

I have a question (my very first here) related to 50/50 beam splitters as used in the Mach-Zehnder interferometers (see for example http://en.wikipedia.org/wiki/Mach-Zehnder_interferometer). Let's concentrate on the input beam splitter: A continuous light beam (the input) is split 50/50, one in the 90deg direction and one in the forward direction of the input beam. Momentum/energy conservation teaches us that deflected part of the beam must exert a (tiny) force on the beam splitter. The forward part of the beam will not. In the single photon (quantum) case, the beam-splitter should get a slight "kick" if the photon is deflected, while the beam splitter is left in its original state if the photon goes straight through. If I got this right so far, we can now increase the energy (and momentum) of the photon and decrease the mass of the beam splitter (now coupled to a sensitive piezo transducer or something) so that a photon "kick" can be recorded if deflected. Now "which way" information is available and interference should be destroyed as I understand this. Would interference be restored if the transducer is disconnected? If yes, how do the photon "know" whether the transducer is connected or not? Maybe this is related to how much the photon/beam splitter system can become entangled to the environment? If the answer is "No", how weak must the photon kick be in order for the interferometer to work? We already know that it works for ordinary light (very tiny photon kicks). Update: Maybe I can reformulate a bit: Why does not the momentum exchange (or lack thereof) between the photon and the beam-splitter (and the trace this leaves in the environment) destroy the interference? Anyway, I see that no-one is interested in this question. Is the question very poor? How can I improve it? If someone just could point me to relevant some literature were transfer of momentum between between photons and optical elements are discussed.. Regards, Carl


Gauge invariance and the form of the Rarita-Schwinger action

in Weinberg Vol. I section 5.9 (in particular p. 251 and surrounding discussion), it is explained that the smallest-dimension field operator for a massless particle of spin-1 takes the form of a field strength, $F_{\mu\nu}$. This is because a massless vector field does not transform properly under Lorentz transformations (Weinberg, Vol. I eq. 5.9.22) and requires additional gauge symmetry/redundancy to 'mod out' unphysical degrees of freedom. One of the questions in Banks' Modern QFT textbook (problem 2.10) asks to repeat this general analysis for spin-3/2 and spin-2 fields, for which we know the main examples are the Rarita-Schwinger field and the graviton. For spin-3/2, however, I am confused as to why the Rarita-Schwinger field is described by an action which looks very similar to the Dirac action. How is it that the massless Rarita-Schwinger field, which carries a vector index as well as an additional spin-1/2 index, is able to avoid the ill-behavior under certain Lorentz transformations (which Weinberg calls $S(\alpha,\beta)$) that is exhibited in the massless vector? Naively I would have wanted to write a spin-3/2 field as a $(1,1/2) = (1/2,1/2)\oplus(1/2,0)$ representation of the Lorentz group so that the vector index can be treated independently of the spinor index. Shouldn't the argument for massless photons carry over and imply that I should have some kind of `field strength' for the spin-3/2 field? I suspect that what I'm missing is something trivial and silly. Thanks!


Orientation of Magnetic Dipoles

Does a magnetic dipole (in a permanent magnet) tend to align with the B-field or with the H-field? The current loop (Ampère) model of the magnetic dipole suggests the former, while the magnetic-monopole (Gilbert) model suggests the latter.


How to express answers where inequalities are involved?

Part of the floor of a workshop is inclined at 10 degrees to the horizontal. This is to allow the safe storage of steel cylinders. A cylinder of mass 7000 kg is stored as shown in the diagram (c the link below). The magnitudes of the forces exerted on the cylinder by the wall and the floor are Rw and Rf newtons respectively. Find the values of Rw and Rf. If the sloping part of the floor was inclined at alpha degrees (0 < alpha < 45) instead of at 10 degrees, show that a) Rw < 70000< Rf, whatever the value of alpha. b) Rw and Rf both increase with alpha. http://www.flickr.com/photos/8625954...in/photostream i did this: R (II to slope) : Rwcos10 = 70000sin10 ---- Rw = 12342.9 N R (perpendicular to slope) : Rf = 70000cos10 + 12342.9sin10 ----- Rf = 71079.9 N I am able to get Rw and Rf on the cylinder. But I don't know how to approach mathematically expressing the second part of the question (the proof part). Parts (a) and (b) of the second part. Could you please help?


Specific electron energy gap values $E_{i+1}-E_i$ vs. photons with arbitrary energy $\hbar \omega$

The energy levels of electrons in an atom are quantized $E_i$. A photon of a specific momentum $\vec p$ and energy $$\omega=(E_{i+1}-E_i)/\hbar$$ hits an atom and gets absorbed. Okay now say the energy of the photon is $$\omega'=(E_{i+1}-E_i)/\hbar+\Delta E/\hbar,$$ where the value $\Delta E$ is some energy, which is not even near the value of any electron state gap energy. hence the quantity $\Delta E$ cannot be absorbed by means of excitation of the other particle. What happens in this case? Does this energy- wise non-fitting photon just go through the material? Does the photon "spit" in a way that the part $E_{i+1}-E_i$ of the photon energy gets absorbed and a new photon with energy $\Delta E$ will move away? Microscopically, has this anything to do with soft radiation? Okay well, if we model the atom as only being able to suck up energy by changing it's energy levels, this would violate momentum conservation. I guess the atom must start to move too, or if we view it as part of a bigger solid state system, the photon will make the thing wobble a little. From that perspective, we'd have to consider the kinetic energy distribution of the particles composing the material and this would be able to suck up any real energy or momentum value. If we leave the microscopic description (in terms of which I'd like to see an answer to the first question - after all, I'm speaking of the behaviour of a single photon here) and go on and view this as a "gray body" problem, and in case the answer really is that after the photoelectric effect the rest energy gets termalized, then I'd like to know the relation (for a class of materials) between the absorption per frequency and material characterizing parameters like (electron/atom density? emissivity? ...).


Hamiltonian of Harmonic Oscillator with Spin Term

We have the usual Hamiltonian for the 1D Harmonic Oscillator: $\hat{H_{0}}=\frac{\hat{P^2}}{2m} + \frac{1}{2}m \omega \hat{X^2}$ Now a new term has been added to the Hamiltonian, $\hat{H} = \hat{H_0} + \mu B\hat{S_z}$ The system has a spin degree of freedom with $s=\frac{1}{2}$ What are the new eigenstates of the Hamiltonian and what are the energy eigenvalues? We have been denoting the stationary states as $|n,s_z\rangle$ and the non-spin eigenvalues as usual $E = h \omega (n+1/2)$ Can anyone give me some help with this question? I'm guessing that the eigenstates don't change and then it's straightforward to find the eigenvalues but I'm not sure if this is correct (or why it would be correct).


Proving that the free energy is extensive

If I have two system of an Ideal gas $A$ and $B$ each of these system has a partition function: $Z_{A,B} = \left ( \frac{V_{A,B}}{\lambda_T} \right )^{N_{A,B}}$ Where: $\lambda_T = \left ( \frac{m}{2\pi\beta \hbar } \right )^{\frac{1}{2}} $ The free energy is: $F_{A,B} = -kT \ln \left ( Z_{A,B} \right ) = -kT N_{A,B}\ln \left (\frac{V_{A,B}}{\lambda_T}\right)$ For the free energy to be extensive the following must be true: $F_{A} + F_B = F_{A+B} \Rightarrow Z_A \cdot Z_B = Z_{A+B}$ However: $Z_A \cdot Z_B = \left (\frac{V_A}{\lambda_T} \right )^{N_A} \left ( \frac{V_B}{\lambda_T} \right)^{N_B}$ and: $Z_{A+B} = \left ( \frac{V_{A+B}}{\lambda_T} \right )^{N_A + N_B}$ So, for $Z_A + Z_B = Z_{A+B}$ to be true $V_A^{N_A} V_{B}^{N_B} = \left(V_A + V_B \right)^{N_A + N_B}$ must be true as well but this isn't true for any system. Since we cannot create energy by mixing two containers of Argon in the same pressure and temperature, something in my understanding is wrong. Where is my fault?


How do particles, such as electrons become visible?

Quantum mechanics says that atoms are invisible - they do not have some specified location, only a probability distribution. So, how can we see them? If there is to be particle-antiparticle annihilation (or other interactions), the particles must have a fixed location, right? So, is this process just random? Is it impossible to know whether a pair will annihilate, but only know the probability?


How does a Trumpet loud speaker work?

I pierced a hole in a cone shaped cardboard's tip and attached it to my phone speaker. Surprisingly the sound produced when attached is three times louder than the sound when it is removed. I do know that it is something relevant to trumpet loud speaker.What makes it to produce louder sounds? ![enter image description here](http://i.stack.imgur.com/VWBpM.png)


Relativity of Simultaniety

![enter image description here](http://i.stack.imgur.com/73vNK.jpg) In the first figure, A and B are two equidistant points from the observer O in S. In the second figure (reference frame S') the corresponding points are A'and B' such that A'O'=O'B', where O' is the observer in S'. The first figure represents the state of S and S' at time t=0 when two events occur at A and B in S and the corresponding points In S' are A' and B'. Now S' moves with a velocity v w.rt. S. The second figure represents the state of S and S' at time t=t1 when the rays from A and B reaches O simultaneously (if possible). Clearly, since at that time O and O' don't coincide, to O' the events should not be simultaneous. Conversely, if at time t=t1' the rays from A' and B' reaches O' simultaneously (if possible), for the same reason O will not find the events to be simultaneous. Since the light rays are unique, they cannot meet more than once. So only one observer should find the the events to be simultaneous. In this case which observer will find the events to be simultaneous and why?


Rotation of object on another object under rotation

First, I would like to know the rotational velocity of disk2 if disk1 turns at $\omega1$. The axis "x" is fixed to the ground and disk1 is allowed to turn around it. Axis "y" is fixed to disk1 and disk2 is allowed to turn around it. For me, disk2 turns around axis "x" and around axis "y" but not in the same manner. I can understand its rotational velocity is $\omega2$ and for me even disk2 turns around axis "x" and because disk1 is turning, this doesn't change its rotational velocity $\omega2$. I'm not sure. Can you explain please ? In fact, what's a rotational velocity in this case for disk2 ? ![enter image description here](http://i.stack.imgur.com/Uo4sT.png) Second, with $\omega1=-\omega2$. A motor is fixed between disk1 and disk2 (in axis "y"), stator on disk1 and rotor on disk2 (idea of Floris, thanks). If a torque from rotor increase $\omega2$ this would say another torque from stator increase $\omega1$. The motor need energy for give a torque to disk2, I'm agree with that. But I don't understand why the motor need energy for apply its torque from stator to disk1. For me, like stator and rotor turn at the same rotational velocity around axis "x" it's not difficult to add torque to disk2 even the stator+rotor turn around "x" in the same time. The motor turn around the axis "x", when stator turn $d\theta$ degree in one direction, rotor turn $d\theta$ degree in the same time. So, it's not "difficult" (need energy) to apply torque on disk1 for me. If rotor moves far away from stator, in this case I'm agree, I will be more difficult to accelerate it, but here stator and rotor never move from each other. If I need to give forces that stator gives to disk1, I'm agree I need energy, but these forces are only a reaction from the stator because rotor apply a torque. I'm not sure I explain very good what I think, I hope someone can clear this idea. I would like to understand why the motor need energy for apply its torque to disk1. \------------------------Added from comment of Floris: I'm agree, if I'm on object that move back and I try to move an object in front of me, I need more energy. I drawn disks and 2 points A and B: ![enter image description here](http://i.stack.imgur.com/OJYLN.png) Imagine, disk2 don't turn around its center of gravity (axis y). Motor and disk2 turn only around axis "x". You could look at distance AB, it is always the same even the stator rotates of 45°, because the rotor rotates of 45° too. It's not like stator move in one direction and rotor in other. Now, the rotor turn around axis "y" but why it's not the same with a linear trajectory ? Now, distance AB change but only because stator "attack" rotor, in the same time stator rotates around axis "x" but rotor rotates around axis "x" too. Maybe my confusion come from I dissociate 2 movements: first the rotation around axis "x" and second the rotation around axis "y". For me, inertia and centrifugal forces rotate motor and disk2 around axis "x" and torque from motor add rotational velocity of disk2 around axis "y", I don't see where the stator move "back" like your example, could you explain more please ? Maybe I don't have the right to think like that. Sorry if my level of physics is low. I added forces, maybe like that it's easier to explain the problem: ## ![enter image description here](http://i.stack.imgur.com/qEhuC.png)


Water falling, why does it spin?

I have noticed that water, when falling will rotate. Looking closely at a thin stream from a faucet and placing a flat object mid stream you will see the water is rotating. The further down the stream you go you will notice the rotation is much quicker. Am I confused at this occurrence or is there a reason? I can only come up with the rotation of the earth taking into affect.


Is it possible to calculate the half-life of a nucleus?

Is it possible to calculate theoretically the half life of a nucleus (how?) or they are all known through measurements? As an example how do you calculate the half life of Carbon-14?


Molecular Hamiltonian

I was reading some material on the Molecular Hamiltonian on Wiki. It said that, > _Almost all calculations of molecular wavefunctions are based on the > separation of the Coulomb Hamiltonian first devised by Born and Oppenheimer. > The nuclear kinetic energy terms are omitted from the Coulomb Hamiltonian > and one considers the remaining Hamiltonian as a Hamiltonian of electrons > only._ I am not sure why this is done. This won't be the "full" Hamiltonian if we omit the KE terms of the nuclei. Can somebody please explain to me why the nuclear kinetic energy terms can be ignored from the Coulomb Hamiltonian.


Relationship between gauss and decibels

In my ongoing effort to understand the world around me, I want to wrap my head around the relationships between two units of measure. Specifically gauss and decibels. The quandary comes from my understand of waves. Hertz is the measure between wave peaks and describes the frequency of the wave. And from what I know about sound, decibels is the measure of intensity. I've played around with a spectrum analyser designed for sound and really got an understanding of how it all works. Then I turned my attention to electromagnetic waves. I was looking for a tool that I could use to measure electromagnetism at any given location. The first tool I came across was an electromagnetic field meter (EMF meter) which is measured in gauss. Wouldn't it be equally effective to measure in decibels? I also could not find a conversion from one to another, so I would assume they measure different things. Can you clear this up for me?


Thermal Energy from Friction on a Rope

Question: A capstan is a rotating drum or cylinder over which a rope or cord slides to provide a great amplification of the rope's tension while keeping both ends free. Since the added tension in the rope is due to friction, the capstan generates thermal energy. If the difference in tension ($T_0-T'$) between the two ends of the rope is $520N$ and the capstan has a diameter of $10.0 cm$ and turns once in $0.90s$, find the rate $P_{thermal}$ at which thermal energy is being generated. My Work: $F = 520N$ $d = 0.1m$ $\Delta T = 0.90s$ $\Delta U = Q - W \implies Q = \Delta U + W$ $W = \tau \Delta \theta = Fd(2\pi) \implies W = 326.7J$ $P = \frac{\Delta W}{\Delta T} = 363W$ However, this is the incorrect answer. (I don't know the correct one.) Can anybody give me any guidance here?


Testing the speed of gravitational waves in superfluids

Ok. Light can be slowed down in superfluids. I would like to know whether if a Cavendish like experiment could be possible within inside superfluids i.e, to rotate two balls such that to observe their gravitational attraction just like the big G is determined and test whether gravitational waves(might take debroglie wavelength) are slowed down or not? Or whether it could be tested for celestial bodies' gravitational wavelength (or debroglie wavelength) redshifted well enough to an observable limit. Would deduce something about gravitons?


Quantum tunneling effect in a potential of the kind $V(x)=A\frac{x^2}{1+x^4}$

Given a potential: $$V(x)=A\frac{x^2}{1+x^4}$$ with $A\gt 1$ and a quantum particle inside the well around the point $x=0$. I'm stuck on the calculation of the transmission and reflection coefficients for this particle vs. its energy.