Immediately after the Big Bang, was the universe in a state of extremely low or extremely high entropy?

Phase space theory suggests that the largest course-graining region, $p$, in a phase space, $P$, is the point in the phase space with the highest entropy. As such, it is in thermal equilibrium with the course-graining regions surrounding it. Now, when an intensity vs frequency graph is obtained for the CMB (Cosmic Microwave Background), a curve almost identical to Planck's curve representing black-body radiation is seen. The temperature of the CMB is observed to be almost uniform across the universe, and the curve of black-body radiation represents a point of thermal equilibrium for a system. Since the CMB has the same curve as black-body radiation, this would suggest that at the birth of the cosmic microwave background, the universe was in a state of thermal equilibrium. Referring back to the phase space spoken of in the first paragraph, since the universe was in a state of thermal equilibrium, it was at a point of highest entropy. However, surely due to the Second Law of Thermodynamics, the entropy of the universe would need to decrease - so it wouldn't be possible for the entropy of the universe to be in such a high state so soon after its birth - so it should surely begin in a low state of entropy. So what's the deal here? Did the universe begin in a really low state of entropy, or a really high one? Are all my explanations here logical or quite inaccurate?


Has everything we see happened in the past?

Whatever we see is basically based on the light that hits our eyes, right? When we look at the Moon we are looking at the Moon as it was couple of seconds ago, as the light takes some seconds (~2 seconds I believe) to reach us. In a way aren't we ALWAYS seeing the past as, no matter how close we are to something, we can never be at distance 0. Am I wrong in thinking about it like that?


Free electrons and energy states

Ok, background - studying for the physics part of my radiology exams, and came across a question that went something like this An electron fired through a tungsten target loses energy by: a) Bremsstrauhlung b) collisions with bound target electrons c) giving off characteristic radiation d) all of the above Now, by the process of elimination d must be right, because both a and b are My question regards c - characteristic radiation My understanding of this process is the excitation of the electron to a higher energy state, usually a more peripheral valence shell, which then releases that extra energy when it drops back (or another drops in to fill the hole) So, what I don't get is how a free electron (like one fired from the filament to the target) has energy states. By definition, it is unbound, and therefore there are no binding energies to create quantised energy states. Is there a property I am overlooking, is the question wrong, or is it a fudge? I apologise if this has been asked. I looked at the suggested answers, and did a google search of stack exchange, but my phone can't use the search bar here directly, so I could have missed something


How is the Saddle point approximation used in physics?

I am trying to understand the saddle point approximation and apply it to a problem I have but the treatments I have seen online are all very mathematical and are not giving me a good qualitative description of the method and why it's used and for what it's used. So my question is, how is the saddle point approximation used in physics? What is it approximating? When can it be used?


Is decoherence even possible in anti de Sitter space? The spatial conformal boundary acts as a repulsive wall, thus turning anti de Sitter space into an eternally closed quantum system. Superpositions remain superpositions and can never decohere. A closed system eventually thermalizes and undergoes Poincare recurrences. Whatever the dual is to an interacting conformal field theory, is it really a theory of quantum gravity, or does it only look like one superficially?


How do voltage and voltage drops over a circuit relate to work done?

I have a few remaining uncertainties when it comes to voltage. I numbered the paragraphs in my thought sequence for easier reference.   1. In a series circuit, I know voltage is the electric potential, and represents the potential for work, but that doesnt mean that any electron travelling on any 10 volt circuit does the same amount of work from start to finish, does it?   2. What i mean is, let's say you have a 10 volt series circuit with a 1 ohm wire. Tracking an electron in the current from start to finish, as work is done on the electron by the voltage, its potential drops, until it reaches 0 at the end of the circuit.   3. But if we increase the resistance of the wire to 5 ohms, it would take 5x longer for the electron to get to the end of the circuit, because the current would be 1/5th, right?   4. But then that means that the electron in the 5 ohm circuit would have done 5x the amount of work (or work done on it) of the 1 ohm circuit over 5x the duration.   5. I know the work done per second is the same, but ultimately, if each electron in the 5 ohm circuit has more work done on it over the course of the circuit, then voltage cant be an expression of total work done from p1 (start of circuit) to p2 (end), right? I think i first thought that because voltage relates to potential for work, and the voltage always drops from full to 0 by the end of the circuit, so i assumed the work must be the same regardless of the circuit resistance.   6. So does this mean that when voltage is divided over the components of the circuit, it's really representing the ratio of where in the circuit the work is done?   7. i.e. in a 12v series circuit with a 3 ohm resistor and a 1 ohm resistor, there would be a voltage drop of 9v over the 3ohm resistor, and 3 volts over the 1 ohm, right? which would mean that 75% of total work done in the circuit was at the 3 ohm resistor?   8. But why is it that the more resistors you have in a series, the less the voltage drops over any one resistor? I know with more resistors the current would be lower. Does voltage drop over a resistor decrease as current decreases?


Darcy law yields extreme speed for gas flow throgh packed spheres?

The darcy law is as follows: $u=-\frac{k}{\mu}\nabla p$. Assume we have a gas, then $\mu$ is about $10^{-5}$. $k$ for packed spheres a few mm in diameter is of order $10^{-8}$ $m^2$. Say the pressure difference across a 0.1 m bed is 10 Pa. Then we get $u= 0.1$ m/s which seems really high for such a small pressure difference. I mean, tens of Pascals is a variation which is comaparable to random pressure fluctations. Is this reasonable?


EM vector potential

We can write the electromagnetic field tensor as $$\begin{bmatrix} 0 & -E_x/c & -E_y/c & -E_z/c \\\ E_x/c & 0 & -B_z & B_y \\\ E_y/c & B_z & 0 & -B_x \\\ E_z/c & -B_y & B_x & 0 \end{bmatrix} = F^{\mu\nu}.$$ Erick J. Weinberg, _Classical Solutions in Quantum Field Theory: Solitons and Instantons in High Energy Physics_ (p. 43), states: > For a static solution with vanishing electric fields $F_{0j}$, and hence > $A_0 = 0$, [...] How can this be proven?


One-photon-at-a-time effect on double slit experiment with detectors?

I do not have a access to a physics lab, nor have I come across this type of a double slit experiment, hence I am curious if someone has tried this. Given a double slit experiment setup with a detector. What happens if we have a detector at one of the slits, but we do not send one photon at a time (in other words, we send a bunch of photons without concerning ourselves about the count)? My assumption is that we will see a particle (bullet) pattern like we would if we sent one photon at a time, is this correct?


Apparent size of objects under water

A friend of mine has a homework question and we're having some trouble figuring out what physical mechanisms come into play for this. > An underwater swimmer sees a spherical air bubble that appears to have a > diameter $d=$ 1.5 cm. What is its actual diameter? We are having trouble, specifically, thinking of physical mechanisms that would change the apparent sizes of objects when seen underwater. Is it caused by refraction at the curved surface of the bubble?


Values of SM parameters at one certain scale

The general question is: What are the values of Standard Model parameters (in the $\bar{MS}$ renormalization scheme) at some scale e.g. $m_{Z}$? As its parametrization in Yukawa matrices is not unique - what are the values of gauge couplings, fermion masses and CKM matrix? The background: I want to solve renormalization group equations of MSSM and in order to have initial conditions for them I need to know SM parameters at one scale - not at few different, which one can find at Particle Data Group webpage.


Acceleration of 2 bodies tied with a string

Find the acceleration of the block of mass M shown in the figure . The co- efficient of friction between the 2 blocks is μ1 and that between the bigger block and ground is μ2. ![enter image description here](http://i.stack.imgur.com/yzma2.png) Could someone help solve this problem using Lagrangian mechanics. I know the solution to this problem using Newtonian mechanics.However I am not finding the same answer when I am trying to do it using Lagrangian mechanics. Answer - $a=\frac{[2m-\mu_2(M+m)]g}{M+m(5-2(\mu_1-\mu_2))}$


Will a ball slide down a lumpy hill over the same path it rolls down the hill?

Suppose I have a lumpy hill. In a first experiment, the hill is frictionless and I let a ball slide down, starting from rest. I watch the path it takes (the time-independent trail it follows). In the next experiment, the hill stays the same shape, but the ball now rolls without slipping down the hill. Assume there is no deformation of the ball or of the hill, no micro-sliding of the contact surface, and no other forms of rolling resistance. Energy is conserved. I release the ball from rest at same point on the hill. I track the path the rolling ball takes. Does the rolling ball follow the same path as the sliding ball, but more slowly, or does it sometimes follow a different path? Note: There may be some "trick" answers. For example, if the hill curves significantly on scales similar to the radius of the ball, the ball could get "wedged in" somewhere, or two parts of the ball could contact the hill at the same time. Let's assume that geometrically, the shape of the hill is such that it is only possible to have the ball contact the hill at one point, and that the ball always contacts the hill at a single point (i.e. it never flies off).


Question about the parity of the ghost number operator in BRST quantization

Given a Lie algebra $[K_i,K_j]=f_{ij}^k K_k$, and ghost fields satisfying the anticommutation relations $\\{c^i,b_j\\}=\delta_j^i$, the ghost number operator is then $U=c^ib_i$ (duplicate indices are summed). The question raised in proving the BRST operator raising the ghost number by 1, given in the Example 6.1 on the page 116-118 of the book, String theory demystified. http://books.google.com/books?id=S4JyPgw4ZlAC&lpg=PP1&pg=PA117#v=onepage&q&f=false On the second and the third lines of the formula derivation of $UC^iK_i$ $UC^iK_i=...=c^iK_i-c^i\displaystyle\sum_rc^rb_rK_i=c^iK_i+c^iK_i\displaystyle\sum_rc^rb_r=...$ where we need $c^r U K_r =-c^r K_r U$ The change of sign above is not manifestly obvious to me. \---add--- Never mind. I find the second line cheated me. It should be $UC^iK_i=...=c^iK_i+c^i\displaystyle\sum_rc^rb_rK_i=c^iK_i+c^iK_i\displaystyle\sum_rc^rb_r=...$ Furthermore, in the proof followed in this example, there a few typos need to be corrected. But the final result is right. The BRST operator raises the ghost number by 1!


degree of freedom of a rigid body 5 or 6?

I'm confused here. I have a three particle (rigid) system. What would be the degree of freedom? I found out five. 3 coordinates for center of mass and 2 for describing orientation.   But we have only three constraints, i.e. three equations that reduce 9 coordinates by 3, 9 - 3 = 6, which gives 6 degrees of freedom?? Did I miss something above?


Information relationship to Special Relativity

How do we write mathematically that "information" cannot go faster than light? And along a similar line of thought, how do we relate "information" with special relativity. Lastly, what is the relationship between Special Relativity and the fact that the phase velocity of a wave packet can go faster than light (light speed here being the group velocity). Is there a reason we cannot consider the frame of reference of a specific phase in a wave packet?


The average value of the the square of Dirac velocity operator

Let's have Dirac velocity operator (the case of the free particle: $$ \hat {\mathbf v} = i [\hat {H}, \hat {\mathbf r}] = \hat {\alpha}, \quad \hat {H} = (\hat {\alpha} \cdot \hat {\mathbf p}) + \hat {\beta} m, $$ where $\hat {\alpha} $ is $\hat {\gamma}_{0}\hat {\gamma}$. I need to find $\sqrt{\langle (\Delta \hat {\mathbf v})^{2}\rangle} = \sqrt{\langle \hat {\mathbf v}^{2}\rangle - \langle \hat {\mathbf v}\rangle^{2}}$. The second summand is equal to $\frac{\mathbf p^{2}}{p_{0}^{2}}$, and the first (maybe) $$ \langle \hat {\mathbf v}^{2}\rangle = \langle \hat {\alpha}^{2} \rangle = 3 \langle \rangle = 3. $$ So $$ \sqrt{\langle (\Delta \hat {\mathbf v})^{2}\rangle} = \sqrt{3 - \frac{\mathbf p^{2}}{p_{0}^{2}}}. $$ But it seems that this result is incorrect, because it is unnatural. Can anyone help me? What is the correct answer?


Color of cementite

I have a question about micrograph of steel. In eutectoid steel, the pearlite exists as grains, within each colony the layers are oriented in essentially the same direction, which varies from one colony to another. The thick light layers are the ferrite phase, and the cementite phase appears as thin lamellae most of which appear dark. However in photomicrographs of a hypereutectoid steel (e.g.1.4 wt% C) , we can observe the proeutectoid cementite appears light. Why there is such a difference?


Pressure derivatives of the bulk modulus of materials? (Solved)

Isotherm bulk modulus is defined by $$K=\rho\left(\frac{\partial p}{\partial\rho}\right)_T=-v\left(\frac{\partial p}{\partial v}\right)_T$$ wherein $v=\tfrac{1}{\rho}$ is the specific volume. Studying around the Birch- Murnaghan equation of state and some other related topics I saw everyone talk about the pressure derivative of the isotherm bulk modulus very happily as if it is a function of e.g. pressure, like any other thermodynamic quantity usually is, so that one can calculate its nontrivial pressure derivative for it, even a nontrivial isotherm pressure derivative $\left(\tfrac{\partial}{\partial p}\right)_T$ of it. See for example this link that introduces the quantity $$K'=\left(\frac{\partial K}{\partial p}\right)_T$$ (the reference cited uses $B$ instead of $K$.) But to me its very odd. Let me compute the isotherm pressure derivative of $K$ so you understand why I'm confused: \begin{align} K'&=\left(\frac{\partial K}{\partial p}\right)_T\\\ &=\left(\frac{\partial}{\partial p}\right)_T \biggl[\rho\left(\frac{\partial p}{\partial\rho}\right)_T\biggr]\\\ &=\left(\frac{\partial\rho}{\partial p}\right)_T\left(\frac{\partial p}{\partial\rho}\right)_T+\rho\left(\frac{\partial^2 p}{\partial\rho\partial p}\right)_T\\\ &=1+0\\\ &=1 \end{align} A constant, not a function of pressure at all. Oh I don't mean that the Bulk modulus is not an ordinary thermodynamic quantity, but its isotherm pressure derivative can not be function of pressure as the above calculations suggest, yet isentropic pressure dependence could be non-trivial and etc. No reference could I have found to have computed this derivative, everyone who I encountered have only reported to have measured the quantity experimentally. That everyone say it is a function of pressure and experiments validate their idea makes me think I have missed a point in my computations. But there is a greater point that suggests i am wrong, that using the specific-volume-based form of the bulk modulus when is differentiated will result in $-1$ instead of $1$: \begin{align} K'&=\left(\frac{\partial K}{\partial p}\right)_T\\\ &=\left(\frac{\partial}{\partial p}\right)_T \biggl[-v\left(\frac{\partial p}{\partial v}\right)_T\biggr]\\\ &=-\left(\frac{\partial v}{\partial p}\right)_T\left(\frac{\partial p}{\partial v}\right)_T-v\left(\frac{\partial^2 p}{\partial v\partial p}\right)_T\\\ &=-1-0\\\ &=-1 \end{align} So that the contradiction $1=-1$ suggests that I must be wrong, but where is the point I am missing? Aren't the derivatives interchangeable in computing $\left(\frac{\partial}{\partial p}\right)_T \left(\frac{\partial p}{\partial\rho}\right)_T$ ( _I understand the Thermodynamic's underlying geometry can be non-euclidean in different cases_ ) or else? So the simple question is this: where have I make a mistake in the above calculations? Regards. _**UPDATE_. Wrong calculation led into an apparent contradiction, but as could be guessed it was wrong so no contradiction at all. It was a math question which is now resolved, so maybe better to be deleted from ere as it had no physics questioned behind it.**


Heuristic argument for the temeprature dependence of specific heat in the "low" temperature regimes

Here by "low temperature" I meant it in the scale of the characteristic $\hbar \omega$ of the system. One can calculate and show that in the low temperature regime $C_V$ of phonons goes like $T^3$ for free fermions it goes as $T$ and for free bosons its like $T^{\frac{3}{2}}$ In most of the above cases the power comes from some complicated calculation.   * I was wondering if there is a heuristic argument for "understanding" these powers.    * Further is the power of $T$ characteristic of the system? Like if in a substance one sees the low temperature behaviour of $C_V$ to be going as $T$ then does it says something like that the effective degree of freedom of the system is that of free fermions?   * Are there other examples of such calculation? Like typical systems where one can calculate the low temperature dependance of $C_V$?


Too small error on the calculus of wavelenght

I have this function: $$\lambda=d \sin(\arctan(\frac{x}{z}))$$ and I want to find its absolute error. $d$ is a constant ($10^{-6}$), $x =(0.716 \pm 0.001)$ m, and $z=(1.000 \pm 0.001) $ m. For the error of $\lambda$ I have used $$\Delta\lambda=\sqrt{\left(\frac{\partial\lambda}{\partial x}\Delta x\right)^2 + \left(\frac{\partial\lambda}{\partial z}\Delta z\right)^2}\\\ \Delta\lambda=d\sqrt{\left(\frac{z\Delta x}{(z^2+x^2)(\sqrt{\frac{x^2}{z^2}+1})}\right)^2 + \left(\frac{x\Delta z}{(z^2+x^2)(\sqrt{\frac{x^2}{z^2}+1})}\right)^2}$$ And I have obtained $6.6 \cdot 10^{-10}$ m. I expect a much larger error. Is this wrong?


conformal, Weyl transformations, apparent discrepancies and confusions

Because of the apparent discrepancy of how some CFT and GR books define conformal transformation unlike in string theory area, I wanted to get rid of all the confusion from McGreevy's lecture notes: > http://ocw.mit.edu/courses/physics/8-821-string-theory-fall-2008/lecture- > notes/lecture10.pdf I have understood and happy with the definition of conformal group and conformal transformation now, but I have the following two questions:   1. In page 2, where he defined conformal invariance, did he mean invariance under the diffeomorphism part only (which gives the conformal group)? The reason I think so is that he changed the metric which according to his definition of conformal transformation don't change. So, the reason I am asking is that I thought conformal invariance will be a suitable name for a theory, invariant under conformal transformation.   2. Just to make sure: the transformation in page 2: $\delta g_{\mu\nu}=2\lambda g_{\mu\nu}$ and $\delta g_{\mu\nu}=\delta\Omega(x)\delta g_{\mu\nu}$, these are results of diffeomorphism ..right? and not Weyl in any way. In other words, what are these transformations? All my questions are for general dimension and general space time (classically of course).


Help in understanding the derivation for Fresnel Distance?

An Aperture of size $a$ illuminated by a parallel beam sends diffracted beam (the central maxima) in angular width approximately $λ/a$. Travelling a distance $z$, it aquires the width $zλ/a$ due to diffraction. The distance at which this width equals the size of the aperture is called the **fresnel distance**. But as far as I know, the angular width of central maxima is given by $2λ/a$ and not $λ/a$. Help?


Why electrons are relativistic in Graphene and non relativistic in vacuum?

If a free region in space has a potential difference of one volt, an electron in this region will acquire kinetic energy of 1 eV. Its speed will be much smaller than the speed of light hence it will be a non relativistic electron. On the other hand conduction electrons in graphene are relativistic for the same potential difference. Question is how come that when the electrons are in vacuum they are non relativistic, and when they are inside Graphene they are relativistic (for the same potential difference)?


Orbital momentum and spin transfer through gauge mediator

I study the jellium model: conduction electrons free except for their mutual repulsion. We have the following vertex: $$\hat{V}_{ee}~=~\sum_{k,k',q}V(q)a^{\dagger}_{k-q,\sigma}a^{\dagger}_{k'+q,\sigma'}a_{k',\sigma'}a_{k,\sigma}. $$ One sees that the electron spins z-projections are unchanged after the interaction $(\sigma,\sigma')\rightarrow(\sigma,\sigma')$ (consequence of the fact that the potential is scalar I guess) so or the mediating gauge boson (photon) isn't carrying any orbital momentum or helicity along the $z$-axis which is unphysical but maybe acceptable for a virtual particle, or there is on orbital momentum conservation at the vertex points. This point is a bit messy in my head, I'd appreciate if someone could share his interpretation or motivate the conservation of each electron's spin state.


Spin 3/2 Statistical Mechanics Problem

I am trying to solve a problem from the book 'Introductory Statistical Mechanics' (Bowley, Sanchez). The question reads: Calculate the free energy of a system of N particles, each with spin 3/2 with one particle per site, given that the levels associated with the four spin states have energies e, 2e, -e, -2e.... What I want to know is how I use the face that each particle has a spin 3/2? Does this add some kind of degeneracy I need to take into account?


How did pre-Copernican astronomers accurately predict planetary position?

Copernican elements (circular orbital elements) are not very accurate. But Copernicus simplified our understanding a great deal by placing the Sun at the center of the system. Im astonished by the historical accounts that, say, Mayan or Babylonians, or whoever may have had the ability to predict planetary position an accuracy that competes even with Kepler. Now, Im only curious about the reasoning and the mathematics behind making such predictions from pre-Copernican understanding. What approach did these ancient people take? I cant seem to find any online reference to this approach.


Hoop rolling inside a circular hole

A hoop of radius $b$ and mass $m$ rolls without slipping within a stationary circular hole of radius $a > b$ and is subject to gravity. Use the generalized coordinates the rotation angle $\phi$ of the hoop and the angular position of the hoop’s center $\theta$. We have the rolling without slipping constraint $$b\phi - a\theta=0.$$ The Lagrangian of the system is $$L=\frac{1}{2}m(a-b)^2\dot{\theta}^2+\frac{1}{2}mb^2\dot{\phi}^2+mg(a-b)\cos\theta.$$ The Euler-Lagrange equations with Lagrange multiplier are $$m(a-b)^2\ddot{\theta}+mg(a-b)\sin\theta=\lambda a, mb^2\ddot{\phi}=-\lambda b$$ Solving for an equation of motion of $\theta$, we have $$(2a^2-2ab+b^2)\ddot{\theta}+g(a-b)\sin\theta=0.$$ My questions are   1. how to find the generalized constraint force that makes the hoop roll without slipping?   2. how to find the constraint force that keeps the hoop’s CM moving on a circular path? ![enter image description here](http://i.stack.imgur.com/JnRCG.jpg)


Historical and philosophical reflexions about the concept of energy?

Perhaps you know the books: Concepts of Mass in Contemporary Physics and Philosophy Concepts of Force: A Study in the Foundations of Dynamics by Max Jammer, which discusses mass and force from a perspective which include much historical as well as epistemological, conceptual and axiomatic elements. Are there any good books which discuss the energy concept in a similar way as Max Jammer discussed for example mass and force?


Reason for the experiment

Why is the independent confirmation by separate experiments such as important part of the development of the idea of the existence of electron wave?


Does zero strain always imply zero stress?

In solid mechanics, can I always assume that if an object undergoes no strain, then no stress is applied to it? I think it's true only because I can't seem to find a counter-example.


Inner Product Spaces

I am trying to reconcile the definition of Inner Product Spaces that I encountered in Mathematics with the one I recently came across in Physics. In particular, if $(,)$ denotes an inner product in the vector space $V$ over $F$:   1. $(u + v, w) = (u, w) + (v, w) \text{ for all } u, v, w \in V$,   2. $(\alpha v, w) = \alpha(v, w) \text{ for all } v, w \in V$ and $\alpha \in F$,   3. $(v, w) = (w, v)^* \text{ for all } v, w \in V$, (* denotes complex conjugation) were some of the properties listed in my mathematics course. In physics, however, the inner product was said to be linear in the second argument and $(v,\sum(\lambda_i w_i)) = \sum(\lambda_i (v,w_i))$ where $v$ and $w_i$ are kets in Hilbert space and $\lambda_i$ are complex numbers. To me, these properties of an Inner Product are not compatible. If the first definition of inner product is correct, then I think $(v,\sum(\lambda_i w_i)) = \sum((\lambda_i)^* (v,w_i))$ where $^*$ denotes complex conjugation.


Jumping on earth versus jumping on the moon

Given the following problem: On the moon the acceleration due to gravity is $g_m = 1.62 m/s^2$. On earth, a person of mass $m = 80 kg$ manages to jump $1.4 m$. Find the height this person will reach when jumping on the moon, if the person is wearing a spacesuit with mass $m = 124 kg$. I am a little bit confused as to whether or not the given information regarding mass is actually needed here at all. Assume that $v_0$ is equal in both jumps, and there is no rotational movement, can't we just use the formula for conservation of mechanical energy? $$\frac{1}{2}m v_{f}^2 + mgh_f = \frac{1}{2}m v_{0}^2 + mgh_0$$ And here we can cancel the mass, $m$ since it appears in all terms. So on earth we will have: $$gh_f = \frac{1}{2} v_{0}^2$$ $$9.8 \cdot 1.4 = \frac{1}{2} v_{0}^2$$ $$v_0 = 5.2 m/s$$ Then on the moon, since we know $v_0$, we can then find $h_f$: $$1.62 h_f = \frac{1}{2} \cdot (5.2)^2$$ $$h_f = 8.3 m$$ Would this not be an acceptable way to solve this? If this is wrong, can anyone please explain why this is wrong conceptually?


Inefficiency Comparison of Car Air Conditioning vs. Open Windows

On a recent long, hot journey in Spain, I was pondering which was the most efficient way of cooling the car. Which of these would be the most effective?   1. Switching on the air conditioning, thereby consuming more fuel   2. Winding down the (both driver and passenger) windows, thus disrupting the airflow around the car and ostensibly creating extra drag Let's assume I'm travelling at 100km/h on level road, in a 1.2 litre Fiat 500. (It's a valid assumption, because it was true). Let's also assume that both methods cool me down equally as well. For what's worth, I went with option 2.


Schwarzschild radius in matrix models

The Schwarzschild radius for 11D BHs is given by $l_{11}(l_{11}m)^{1/8}$, which is the special case ($D=11$) of the general dimensional case of $(G_Dm)^{\frac{1}{D-3}}$. Here $m$ is the BH mass and $G_D$ is $D$ dimensional Newton's constant. Now if M-theory is compactified on a torus $T^p$ of sizes $L$s, then I think $G_D$ is related to eleven dimensional Planck length $l_{11}$ as $G_D=\frac{l_{11}^9}{L^p}$. I don't understand how can then the radius smoothly go to the 11 dimensional value if we start decompactifying the torus to get to the 11D case. It seems that it blows up. Where's my mistake? Thanks a lot.


How to prove Gegenbauer's addition theorem?

I asked this at: http://math.stackexchange.com/questions/210153/, but didn't get any reply, so I am trying here, since I actually need this in physics anyway. How can one prove the following identity: $$ V_k(r_1, r_2) = {2k+1\over 2 r_1 r_2}\int_{|r_1 - r_2|}^{r_1+r_2} e^{-{r\over D}} P_k\left(r_1^2 - r^2 + r_2^2 \over 2 r_1 r_2 \right) d r= $$ $$ =(2k+1) {I_{k+{1\over2}}\left(r_<\over D\right) \over \sqrt{r_<}} {K_{k+{1\over2}}\left(r_>\over D\right) \over \sqrt{r_>}} $$ ? Here $r_<=\min(r_1, r_2)$, $r_>=\max(r_1, r_2)$, $I_\nu$ and $K_\nu$ are the modified Bessel functions of the first and second kind, $P_k(x)$ are Legendre polynomials and $k=0, 1, 2, 3, \dots$. I verified the identity numerically, so I know that it works, but I didn't figure out how one could prove it. I know it has something to do with Gegenbauer's addition theorem and the equation 10.23.8 at [1]. The $V_k(r_1, r_2)$ is just the coefficient in the Legendre expansion of: $$ {e^{{|{\bf r}_1 - {\bf r}_2|}\over D}\over |{\bf r}_1 - {\bf r}_2|} =\sum_{k=0}^\infty V_k(r_1, r_2) P_k(\cos\theta) $$ Note: when $D\to\infty$, then $V_k$ becomes ${r_<^k \over r_>^{k+1}}$, which follows from the properties of the Bessel functions and this is just the well known multipole expansion [2]. [1] http://dlmf.nist.gov/10.23#E8 [2] http://en.wikipedia.org/wiki/Legendre_polynomials#Applications_of_Legendre_polynomials_in_physics


Did the Goudsmit-Uhlenbeck analysis of spin consider relativity?

It's frequently mentioned in introductory quantum mechanics texts that Goudsmit and Uhlenbeck conjectured that the magnetic moment of an electron was due to angular momentum arising from the electron rotating around its own axis. But then when they tried to calculate how fast it would have to be spinning, assuming that the electron is a rigid sphere with radius equal to the classical electron radius, they found that a point on the equator would be moving with a speed greater than the speed of light, so they were embarrassed for publishing their work. My question is, did they do this calculation using Newtonian mechanics or special relativity? If we do take relativity into account, and consider a (Born-) rigid sphere with radius equal to the classical electron radius, and then we tried to find out what speed the sphere would need to rotate at in order to have an angular momentum that produces the magnetic moment of an electron, would we still get a speed faster than light? Momentum goes to infinity as speed approaches c, but what happens to angular momentum? I'm aware that angular momentum becomes really complicated in special relativity, with tensors and bivectors and the like, but is there a simple (or even approximate) expression that can give us some idea of what would happen in this case? This is of course just a curiosity, because there are other problems with the classical theory of spin, like the fact that a rotation of 720 degrees is required (for an electron) rather than a rotation of 360 to get you back to your initial state, due to the double cover property of SU(2). Any help would be greatly appreciated. Thank You in Advance.


Stability of D-Branes and Coupling to Fields

I've been brushing up on my string theory using Becker, Becker, Schwarz (p. 208 in particular). They state that some D-branes can couple to R-R fields from closed string excitations. They then go on to say that these branes are exactly 1/2 BPS and therefore stable. I understand that 1/2 BPS objects are stable because they preserve enough of the supersymmetry. But I don't understand why the coupling to R-R fields ensures that the branes are 1/2 BPS. Could someone possibly explain this point mathematically, and/or direct me to a reference that makes it clear? Many thanks for your help! **Edit** : perhaps the answer is contained within Polchinski's seminal paper here? Still might be useful to get a second opinion!


What are the conditions around Sgr A* for a space ship trying to plunge into the central black hole?

The possible problems for a space ship near an active galactic nucleus are the following: tidal forces, hot plasma accreting into the black hole and "jetting" out, radiation, and magnetic fields. If I am missing something, please feel welcome to tell me. * * * **Tidal forces** Suppose we have a ship which has dimensions $< 1\, \rm km$, then tidal forces would surely not pose a problem as this length is much smaller than the Schwarzschild radius of Sgr A* which is presumed to be $r_s \approx 10^7 \rm km$. (You can also verify the negligibility explicitly through the equation of geodesic deviation in a Kerr metric.) * * * **Plasma** With plasma, I am not entirely sure. The accretion disc has many various models including "slim" and "thin" versions, which have a negligible extent beyond the equatorial plane, but also "thick" ones. Nevertheless, even with the "thick" Polish donut models, we see that the main density is concentrated around the equatorial plane (see image below) and there might be pathways passing through reasonably low densities of plasma. On the other hand, there surely is a means of transport of the plasma towards the axis to create jets. So for a perfectly axially symmetric situation, the horizon should be "screened" by a considerable flow of plasma from the disc towards the jets. On the "other other" hand, the typical length-scale of the accretion disc-hole system are proportional to the Schwarzschild radius $ \sim 10^7\rm km$ and this is also the typical length of the accuracy of the models we should expect. For the spaceship to safely pass, we would need an underdensity say of the lenght-scale $\sim100 \, \rm km$ i.e. of order $10^{-5}$ with comparison to the length-scale of the system. Even though a mere "bubble" in a plasma would surely not guide the space ship all the way towards the black hole, it provides a possibility to cross thin regions where the model predicts the plasma to be denser. ![thick disk simulation](http://i.stack.imgur.com/ybDqo.jpg) * * * **Radiation** We know Sgr A* to be a source of radiation all across the spectrum including X-rays. However, we see the radiation already vastly shielded by dust and factors such as the inclination of the disc with respect to us also play a role. Hence, estimating the actual intensity is not straightforward. If the radiation is too strong, the ship gets heated up beyond sustainability, not to mention the damage done on the pilots by the not-shielded X-rays. Once again, the jets can change the game - for sufficiently rapid jets, the radiation will gain much higher energies via inverse Compton scattering thus posing a much larger threat for the spaceship. Once again, I am not able to quantitatively assess the importance of such an effect especially due to the fact that from e.g. this article it would seem that Sgr A* is very much gamma- ray quiet. (Which does not mean the ship and it's crew wouldn't get fried by the "weak" gamma ray production from the vicinity of the jet.) Nevertheless, we should not forget that the ship would still be very close to a black hole and the radiation is not homogenous/isotropic. The strongest radiation comes from the parts closest to the inner edge of the disc at $\sim 3 r_s$. If the pathway was taken away from the equatorial plane, much of the radiation could be devoured or deflected by the black hole's shadow. * * * **Magnetic fields** Accretion is believed to be accompanied by _very_ strong magnetic fields. The two effects which could cause a problem are quickly varying fields causing electromagnetic induction on the electronic devices on the ship, and the effects of a very strong magnetic field on e.g. ferromagnetic parts of the spaceship. The variance of the field could be induced either by the ship moving through a space-varying field or by the inherent time-variance of the field itself. Once again, the variations of the field will be modeled on idealized models with length- and time-scales appropriate for the system itself, not the time- and length-scales relevant for the moving ship. There could be a "noise" in the field either spatial or temporal (on scales $\ll r_s$ or $\ll r_s/c$) strong enough to cause critical damage to the ship through induction. * * * It would be a too broad question to theorize whether a ship could be able to pass through such conditions (I might ask this on Space.SE using the answers to this question). For now, I will ask only about the _conditions_ relevant for a space ship approaching the Black-hole horizon. **QUESTION 1:** What are the quantitative observational constraints on the density and temperature of plasma around Sgr A* and its spatial configuration? Is there a very-low-plasma-density pathway into the black hole allowed by the current knowledge? **QUESTION 2:** What is the intensity of different types of radiation in the very vicinity of Sgr A*? What would be the optimal angle to approach the presumed black hole in it's center to maximally evade the radiation? **QUESTION 3:** What is the presumed structure of the magnetic field in Sgr A*, it's strength and derivatives in both time and space? Is there a possibility of a "strong noise" in the magnetic field say at the time-scales $10^{-4}-10 \, \rm s$ and length-scales larger than $1\, \rm cm$? For the last question we do not know what the viable velocity of the space- ship would be, but assume at least $100 \, \rm m \cdot s^{-1}$. For the moment, let us neglect the fact that the proper time of the background (defined by the quasi-exact Killing vector of the close-to-Kerr space-time) and the space-ship time might differ notably and always describe the situation from the point of view of an observer at spatial infinity. * * * This question is a followup trying to clear up the discussion of this question concerning itself with a spaceship plunging into an (im)possible worm-hole. EDIT: I have added a note on the gamma rays produced by the black hole as suggested by Kyle Kanos.


internal conversion and the electromagnetic force

I have read that the mechanism behind internal conversion, in which a nuclear transition leads to the ejection of an electron in one of the lower atomic orbitals, is related to the fact that the wavefunctions of some of these electrons penetrate the nucleus. Because of this, there is a finite probability that the electron will be in the nucleus, and, when this occurs, the electron may couple to the ~~excited state of the nucleus~~ nuclear transition when it happens. Is it conjecture that the electron must penetrate the nucleus in order for internal conversion to occur, or is this a straightforward result? I would have figured that since internal conversion is mediated by the electromagnetic force (which has long reach), for any electron within range of a virtual photon there will be a nonzero cross section for this type of decay. What details am I missing in thinking this?


Is general covariance a symmetry?

Is general covariance a symmetry? If it is ,what is its symmetry group and corresponding generator?


What happens if object is thrown in empty space?

If I throw a object in empty space, I apply a force to throw that. Then it gains some acceleration and it's speed increases. So will it's speed keep on increasing, or it will get stable? If yes, what happened to the force I applied. No change in speed, so acceleration = 0, then $F = 0$.


If an area remains at temperature 0 Kelvin would the area be frozen in time?

If an area is at 0 kelvin, I am guessing that there is no motion. Without motion in an area, will it not look as if time has stopped?


Conjugate momentum is not gauge invariant

The conjugate momentum of a charged particle moving in a uniform magnetic field is given by $$\vec p=m\vec v+q \vec A$$ This expression is not unique because $\vec A$ is not unique. $\vec A$ is not a measurable quantity. But $\vec p$ is a phase space variable and if it is not unique the prediction of future of the system is not unique either. Isn't this a problem?


What is cp/cv ratio?

Recently in our chemistry class we read that the $C_p/C_v$ ratio of the inert gases is 1.66 thus they show inert nature. I asked my teacher what was $C_p/C_v$ ratio but he didn't answer me. So I want to know what the $C_p/C_v$ ratio really is?


Optical Waveguides with grating

I have some problems about mode analysis in grating-coupled waveguides,Would you please introduce me a useful book about wave propagation in these structures.


Relativistic Effects Observed with Time Travel

Disclaimer: My domain knowledge on these topics is pretty minimal. I'm a "physics fan". From what I understand about relativity. If there are two identical objects, A and B, and A is stationary and B is not. Then A will observe that B's time is elapsing more slowly than its own. It will also notice that B is shorter in the direction of travel than A is. Now lets assume that A and B could somehow see through time as well as space (this is potentially where the question breaks down). I suppose they would see something like a series of cubes, representing each instance in time, with all the objects in different positions. If both A and B were now stationary in space, but B was traveling twice as fast through time than A was. Would there be any relativistic effects A would see as it is observing B?


Equivalent resistance in ladder circuit

I have stumbled upon a given question I really have a hard time to solve. Basically I need to find an equivalent resistance in some form of "ladder" configuration. Where the chain is an infinite sequence of resistors. ![ladder](http://i.stack.imgur.com/l7du7.png) I have really no good idea how to find this equivalent resistance. Trying the old fashioned rule of parallel and resistors in series I came to a very messy formula: I have really no good idea how to find this equivalent resistance. Trying the old fashioned rule of parallel and resistors in series I came to a very messy formula: $$R + \frac{1}{\frac{1}{R} + \frac{1}{R+\frac{1}{\frac{1}{R}+....}}}$$ I know however that the solution should be much more simple. Now I tried using Kirchhoff's loop rule. Which states that the power difference in a closed loop must be 0. Naming the "potential currents" between AB $I_1$, between BC (through the single resistance) $I_2$ and the current "from B to the right" $I_3$ Considering the loop containing BC & the rest of the structure this rewrites to: $$I_1 = I_2 + I_3$$ $$I_3 \cdot R - I_2 \cdot R_{eq} = 0$$ The problem is, 2 variables, 2 functions doesn't really bring me closer to an answer :(. What am I missing?


Why $SU(3)$ and not $U(3)$?

Is there a good reason not to pick $U(3)$ as the colour group? Is there any experiment or intrinsic reason that would ruled out $U(3)$ as colour group instead?


How to show that a general Galilean transformation in three dimensions is a conformal transformation?

Suppose two particles with equal mass move and then collide. We can easily show that the angle of collision is ninety degree if we choose our frame of reference moving with velocity equal to one of the particles velocity. How can we prove that any inertial frame of reference preserve the angle of collision? In other words, how to show that a general Galilean transformation in 4-dimensional space-time is a conformal transformation?


Wavefunction collapse in relativity

It is well accepted that quantum theory has well adapted itself to the requirements of special relativity. Quantum field theories are perfect examples of this peaceful coexistence. However I sometimes tend to feel little uneasy about some aspects. Consider an EPR pair of particles light years apart. Suppose there are 2 observers moving relative to each other with constant relative velocity. Let us consider, there are spin detection mechanism at both end for each particle. Now suppose one of the observer is at rest w.r.t. the detector for the first particle. As soon as the detection made, the wave function of the 2 particle entangled system will collapse instantaneously and the second particle must realize a definite opposite spin value. Now due to relativity of simultaneity, the second observer may claim that the collapse of the wave function for the two particle system is not simultaneous. He may even claim that the second particle is measured first. In that case a special frame of reference will be privileged, the frame at which the wave function collapsed instantaneously. This will cause a significant strain on the core principle of special relativity. I am sure the above reasoning is flawed. My question is where?


How to solve Boltzmann equation using monte carlo methods?

I'm trying to solve for electron and hole distribution function using Boltzmann equation with various scattering mechanisms. Since I land up with an integro-differential equation, analytical soln. is very difficult. So, I need to understand how this problem could be solved using numerical methods. Please help. (Try suggesting some references with illustrative examples which could be useful) Thanks!


Field of a parallel plate capacitor

Pretty much in all the diagrams I have seen the electric field of the parallel plates is depicted as in this image ![enter image description here](http://i.stack.imgur.com/YyLco.jpg) The field lines seems to start bending only after leaving the periphery of the plates, shouldn't the field lines be depicted as starting to bend as soon as we start to move away from the exact centre of the capacitor ?


Spilling a cup of coffee

I've observed this many times when I spill coffee, It gets attracted towards the cup or the vessel. why is this so? Which is the force acting? (Hope you understood my question).


When the matter is heated does the photon absorbed by electrons of atoms or by atoms themself?

When the matter is heated does the photon absorbs by electrons of atoms or by atoms themself?


Perfectly centered break of a perfectly aligned pool ball rack

Imagine the beginning of a game of pool, you have 16 balls, 15 of them in a triangle <| and 1 of them being the cue ball off to the left of that triangle. Imagine that the rack (the 15 balls in a triangle) has every ball equally spaced apart and all balls touching all other appropriate balls. All balls are perfectly round. Now, imagine that the cue ball was hit along a friction less surface at the center axis for this triangle O-------<| and hits the far left ball of the rack dead center on this axis. How would the rack react? I would imagine this would be an extension of newtons cradle and only the 5 balls on the far end would be the candidates to move at all. But in what way would they move? Which ball, for instance, would move the furthest? Thanks


GPS Satellite - Special Relativity

I'm going through an old relativity assignment, and I've been asked to calculate the time dilation for a satellite which orbits the earth in 12 hours at 26000km from the surface, and travels at a constant speed. The given radius of the Earth is 6400km. In the solutions, the author has calculated the velocity traveled by the satellite to be $\dfrac{3.24\times 10^7\times 2\pi}{12\times 3600} \text{ms}^{-1}\simeq 4500 \text{ms}^{-1} $. I'm fine with the calculation (if I've gone wrong with the numbers somewhere, I don't mind). What I'm not so OK with is that they proceeded to use $v=4500$ in the formula $\gamma=\dfrac{1}{\sqrt{1-\frac{v^2}{c^2}}}$, which is needed to calculate time dilation. As far as I am aware, this formula only holds for observers whose relative velocity is $v$. However, in this case where there is an observer on the Earth's surface and a satellite moving in circular motion around the center of the earth, their relative velocity isn't constant!


How to calculate a collision which is partly elastic and partly inelastic?

_(For the purpose of this question, "calculating a collision" means: given the velocities and masses of two objects in a collision, figuring out the new velocities of both objects after the collision)._ I know how to calculate a totally elastic collision, and how to calculate a totally inelastic collision. But I don't know how to calculate a collision which is part elastic and part inelastic. I don't know where to start. Guidance will be appreciated. (Go easy on the math please).


Can quantum entanglement be a proof of a 4th spatial dimension?

What I know about quantum entanglement is almost nothing. What I know can be resumed to: "two particles created together have a correlated behaviour no matter the distance between them". This is often interpreted as a faster-than-light phenomenon because it is instantaneous. But let's consider a 4th spatial dimension (we then have x,y,z,w)   * 2 particles are created in the point x=0,y=0,z=0,w=0   * Those 2 particles move apart in the 3D space such as: Particle 1 is @ x1,y1,z1,w and Particle 2 is @ x2,y2,z2,w Note that they are at the same w coordinate. Can this be an explanation for the fact that the correlation is instantaneous? I am not considering that this is a revolutionary thing. I am sure that a lot o people already thought about this, but I want to understand it better.


How would the night sky appear at the edge of the galaxy?

In Isaac Asimov's Foundation Series, there is a planet named Terminus which is believed to be the planet farthest from the galactic center. > There are almost no visible stars in its sky, only the huge lens of the > Milky Way galaxy But what would we really see if we look at the opposite side from the galactic center (without a telescope or binoculars)? Would we see a black sky or a sky full of galaxies?


Choice of X-ray scatterer in Compton effect

I am going to perform an experiment on Compton Scattering, and I am going to use an X-ray scatterer to scatter the incident X-rays. I have been instructed that Acrylic Glass slab are the best for this purpose. I am not sure if it is because the acrylic glass slab does not have any free electrons present or any other reason. Please explain.


Electric potential of a spheroidal gaussian

I'm looking for results that compute the electrostatic potential due to a spheroidal gaussian distribution. Specifically, I'm looking for solutions of equations of the form $$ \nabla^2\Phi=N\exp\left({-\rho^2/2\sigma_r^2}\right)\exp\left({-z^2/2\sigma_\ell^2}\right), $$ where $\rho^2=x^2+y^2$, possibly with some polynomial factor of the form $\rho^{|m|}\cos(m\phi)z^n$ in front. This has an easy solution when the two variances $\sigma_r$ and $\sigma_\ell$ are equal, in which case there is spherical symmetry applies and Gauss's law readily yield $\Phi$ in terms of error functions. However, I'm having trouble finding results for the general case. Using spheroidal coordinates doesn't seem to help, as the constant-$\mu$ surfaces are confocal spheroids that taper out to spheres as they get larger, instead of maintaining their ellipticity as the charge density does. This looks standard enough that it ought to have been done before (right?), but it is also messy enough that I'm unsure it made it to any textbook. Has anyone seen the likes of this before?


What is intrinsic gravitational entropy? Does it have to do with dark matter or coarse graining in the universe? Is it unique to general relativity, or there are predictions from quantum mechanics as well? Please explain in detail.


Tritium decay is spontaneous even if the binding energy of tritium is higher than the binding energy of 3He. Why?

Given this nuclear reaction: $^3_1\mathrm H\to {}^3_2\mathrm{He}+e^-+\bar{\nu}$ and knowing the binding energies: $BE(^3_1\mathrm H)=8.48 \,\mathrm{MeV}$ $BE(^3_2\mathrm{He})=7.72 \,\mathrm{MeV}$ If I calculate the mass defect (obviously considering the binding energies in the mass calculation) I obtain a positive value: $M(^3_1\mathrm H)c^2=2809.08 \,\mathrm{MeV} > M( ^3_2\mathrm{He})c^2+M(e^-)c^2=2808.991 \,\mathrm{MeV}$ as expected for a spontaneous decay. Considering the binding energies I have written above I expect the $^3\mathrm{H}$ to be more stable than $^3_2\mathrm{He}$. My question is: why does this decay occur?


Is there any anti-gravity material?

I want to know if there is any anti-gravity material. I am thinking of making flying vehicles which are made up of anti-gravity material so that they will not experience any gravity on them and can easily take off and be more fuel efficient. Is there any such thing? Or any workaround?


Does the transmission axis matter for sending polarized light through polarized glass?

If I have polarized light and I send through only one polarized glass plane, does the transmission axis matter, or will the intensity be halved no matter what.


Variational Derivation of Schrodinger Equation

In reading Weinstock's _Calculus of Variations_ , on pages 261 - 262 he explains how Schrodinger apparently first derived the Schrodinger equation from variational principles. Unfortunately I don't think page 262 is showing so I'll explain the gist of it: "In his initial paper" he considers the reduced Hamilton-Jacobi equation $$\frac{1}{2m}\left[\left(\frac{\partial S}{\partial x}\right)^2 \ + \ \left(\frac{\partial S}{\partial y}\right)^2 \ + \ \left(\frac{\partial S}{\partial z}\right)^2\right] \ + \ V(x,y,z) \ - \ E \ = \ 0$$ for a single particle of mass $m$ in an arbitrary force field described by a potential $V = V(x,y,z)$. With a change of variables $S \ = \ K\log(\Psi)$, (where $K$ will turn out to be $\hbar=h/2\pi$) it reduces to $$\frac{K^2}{2m}\left[\left(\frac{\partial \Psi}{\partial x}\right)^2 \ + \ \left(\frac{\partial \Psi}{\partial y}\right)^2 \ + \ \left(\frac{\partial \Psi}{\partial z}\right)^2\right] \ + \ (V \ - \ E)\Psi^2 \ = 0.$$ Now instead of solving this he, randomly from my point of view, chose to integrate over space $$I = \iiint_\mathcal{V}\left(\frac{K^2}{2m}\left[\left(\frac{\partial \Psi}{\partial x}\right)^2 \ + \ \left(\frac{\partial \Psi}{\partial y}\right)^2 \ + \ \left(\frac{\partial \Psi}{\partial z}\right)^2\right] + (V - E)\Psi^2\right)\,\mathrm{d}x\,\mathrm{d}y\,\mathrm{d}z$$ then Weinstock extremizes this integral which gives us the Schrodinger equation. Apparently as the book then claims on page 264 it is only after this derivation that he sought to connect his idea's to deBroglie's wave-particle duality. Thus I have three questions, 1) What is the justification for Feynman's famous quote: > _Where did we get that [Schrödinger's equation] from? It's not possible to > derive it from anything you know. It came out of the mind of Schrödinger._ > The Feynman Lectures on Physics in light of the above derivation. I note that all the derivations I've seen of the Schrodinger equation doing something like using operators such as $i\hbar\partial/\partial t = E$ to derive it always mention it's merely heuristic, yet what Schrodinger apparently originally did seems like a roundabout way of solving the Hamilton-Jacobi equation with no heuristic-ness in sight. What subtleties am I missing here? Why would I be a fool to arrogantly 'correct' someone who says Schrodinger is not derivable from anything you know? Note I've read a ton of the threads on this topic on this forum & none even go near the calculus of variations, in fact the above contradicts this explanation here that even refers to Schrodinger's Nobel prize lecture, so hopefully it's not a duplicate. 2) Is the mathematical trick Schrodinger has used something you can use to solve problems? 3) Why can't you use this exact derivation in the relativistic case? **Edit:** On the topic of complex numbers: On page 276, 14 pages after he explains what I posted, & after he goes through how Schrodinger linked his work with DeBroglie's work, it is only then that Weinstock says: > _In a more complete study of quantum mechanics than the present one the > admissibility of complex eigenfunctions $\Psi$ is generally shown to be > necessary. If $\Psi$ is complex, the quantity $|\Psi|^2$ is employed as the > position probability-density function inasmuch as $\Psi^2$ is not restricted > to real nonnegative values._ Apparently Schrodinger was able to do what I have posted using real-valued functions & have K as I have defined it, without i. If you're following what Weinstock is saying he shows how the hydrogen atom energy levels are explainable without complex numbers, i.e. he is able to derive a physical interpretation of the eigenvalues (discrete energy levels) of the Schrodinger equation that were in accord with experiment (see Section 11.3 Page 279 on). As far as I understand it it is in trying to find a physical interpretation of the eigenfunctions that one is forced into complex numbers, though apparently, according to the book, it can be shown to be necessary. That is something to take into account!


Coffee cooling in a cup

Will coffee in two differant size styrofoam cups in the same room conditions cool at the same rate? One cup is 8 oz. and the other would be 12 oz.


Proof of S-duality between Type IIB, IIB and Type HO, I string theories

About every source on string theory I've read which do mention S-duality state that: $$\begin{array}{l} \operatorname S:\operatorname{IIB} \leftrightarrow \operatorname{IIB}\\\ \operatorname S:\operatorname{HO} \leftrightarrow \operatorname{I} \end{array}$$ However, how does one prove that the Type IIB string theory is self-S-dual and even more bizzarely, that the Type HO string theory is S-dual to the Type I string theory?


Shouldn't the addition of angular momentum be commutative?

I have angular momenta $S=\frac{1}{2}$ for spin, and $I=\frac{1}{2}$ for nuclear angular momentum, which I want to add using the Clebsch-Gordan basis, so the conversion looks like: $$ \begin{align} \lvert 1,1\rangle &= \lvert\bigl(\tfrac{1}{2}\tfrac{1}{2}\bigr)\tfrac{1}{2}\tfrac{1}{2}\rangle,\tag{4.21a} \\\ \lvert 1,0\rangle &= \frac{1}{\sqrt{2}}\biggl(\lvert\bigl(\tfrac{1}{2}\tfrac{1}{2}\bigr)\tfrac{1}{2},-\tfrac{1}{2}\rangle + \lvert\bigl(\tfrac{1}{2}\tfrac{1}{2}\bigr),-\tfrac{1}{2}\tfrac{1}{2}\rangle\biggr),\tag{4.21b} \\\ \lvert 1,-1\rangle &= \lvert\bigl(\tfrac{1}{2}\tfrac{1}{2}\bigr),-\tfrac{1}{2},-\tfrac{1}{2}\rangle,\tag{4.21c} \\\ \lvert 0,0\rangle &= \frac{1}{\sqrt{2}}\biggl(\lvert\bigl(\tfrac{1}{2}\tfrac{1}{2}\bigr)\tfrac{1}{2},-\tfrac{1}{2}\rangle - \lvert\bigl(\tfrac{1}{2}\tfrac{1}{2}\bigr),-\tfrac{1}{2}\tfrac{1}{2}\rangle\biggr),\tag{4.21d} \end{align} $$ where $F=I+S$, so this is the basis $\lvert F m_F \rangle = \sum_m \lvert\bigl(I S\bigr),m_I m_S\rangle $. Now since adding angular momenta is commutative, the exchange between $I$ and $S$ shouldn't mathematically introduce any kind of difference. In other words, in the basis described in those equations, it shouldn't matter whether I write it as $\lvert\bigl(I S\bigr),m_I m_S\rangle$ or $\lvert\bigl(S I\bigr),m_S m_I\rangle$, right? Now the problem is the following: I have created the Hamiltonian matrix $H=-\vec{\mu}\cdot \vec{B} = -2 \mu B_z S_z/\hbar$ in the $\lvert F m_F \rangle$ representation, and actually the result depends on how you call those angular momenta, so the result could be $$H = \begin{pmatrix} \mu_B B & 0 & 0 & 0 \\\ 0 & \- \mu_B B & 0 & 0 \\\ 0 & 0 & 0 &\mu_B B \\\ 0 & 0 & \mu_B B & 0 \end{pmatrix}$$ Or could be $$H = \begin{pmatrix} \mu_B B & 0 & 0 & 0 \\\ 0 & \- \mu_B B & 0 & 0 \\\ 0 & 0 & 0 &-\mu_B B \\\ 0 & 0 & -\mu_B B & 0 \end{pmatrix}$$ Depending on how you "label" them, $I$ or $S$... which is very confusing! This happens because the off-diagonal terms $$\left\langle 1 0 \right| S_z \left| 0 0 \right\rangle = \frac{1}{2} \left( \left\langle (\frac{1}{2} \frac{1}{2}) \frac{1}{2} -\frac{1}{2} \right| + \left\langle (\frac{1}{2} \frac{1}{2}) -\frac{1}{2} \frac{1}{2} \right| \right) S_z \left( \left| (\frac{1}{2} \frac{1}{2}) \frac{1}{2} -\frac{1}{2} \right\rangle - \left| (\frac{1}{2} \frac{1}{2}) -\frac{1}{2} \frac{1}{2} \right\rangle \right)$$ will be either $\hbar/2$ or $-\hbar/2$ depending on your convention whether it's $\lvert\bigl(I S\bigr),m_I m_S\rangle$ or $\lvert\bigl(S I\bigr),m_S m_I\rangle$. How can I understand this physically and mathematically? Shouldn't the addition be commutative and the process be blind to which labels I use?


Problem in a truss

![enter image description here](http://i.stack.imgur.com/9jFoH.png) For the truss above known are the following: l= 1 + 0.02*n [m], h = 2 + 0.01*n [m], F_1 = 200 + 2*n [N], F_2= 500 + 6*n [N], F3 = 400 + 5*n [N], Young's modulus E = 2*10^11 [N/m2], allowable stress σ_a= 8*10^8 [N/m2] In the amounts above, n a natural number. Find:   1. Reaction forces at A and E .   2. Tension forces in all members of the truss. Draw free body diagrams for all joints and members; indicate extension or compression for the member forces.   3. Diameters of all the members considering they all have circular cross-sections. The diameters have to be selected from the following sequence of diameters (in mm): 2, 5, 8, 10, 12, 16, 20, 24, 30, 36, 40, 50, 60, 70.   4. The axial deformation (Δl) of bars BC and FG.


Why is there a 90˚ phase angle between particle velocity and sound pressure in spherical waves?

My text says that in a plane sound wave (or in the far field), particle velocity and pressure is in phase. As we move closer to the sound source (to near field and more spherical waves), the phase angle between these two quantities will gradually shift towards 90˚. Why does this phase shift occur? I haven't been able to find an intuitive explanation for why there is a phase shift in spherical waves, but not in plane waves, and vice versa.


Focal length, power and magnification

In refracting (lens not mirror based) telescopes, to have a large magnification the objective lens needs to have a large focal length. if $\text{Power} = \frac{1}{\text{focal length}}$ then that means telescopes that have a large magnication have a low powered objective lens? Also if a large focal length provides a greater magnification then a lens which refracts (bends) the light the least is better, so surely that's no lens at all (or just a pane of glass)? How does a longer focal length (and so low power lens) provide a greater magnitude? It seems to my common sense (obviously wrong) that a lens that bends the light more would provide a greater magnification? (As the rays would be spread over a greater area for a shorter length)


Question about tide

Let's say a canal is built in the middle of Kansas in the shape of a ring that stretches 100 miles in diameter. The band of water is 100 feet wide and 10 feet deep throughout the entire canal, and the land where the canal is built is flat in every way aside from conforming to the curvature of the earth. Would a current develop from tidal forces? If so, in what direction would it flow? How rapid would it be?


Riemann curvature tensor symmetries confusion

In the context of spacetime, reading Schutz, I'm confused about the symmetries of the Riemann curvature tensor, which I understand are: $$R_{\alpha\beta\gamma\mu}=-R_{\beta\alpha\gamma\mu}=-R_{\alpha\beta\mu\gamma}=R_{\gamma\mu\alpha\beta}.$$ But using the metric to contract the Riemann tensor can't I also say $$R_{\gamma\mu}=g^{\alpha\beta}R_{\alpha\beta\gamma\mu}=g^{\alpha\beta}R_{\alpha\gamma\beta\mu}?$$ Which leads me to think that $$R_{\alpha\beta\gamma\mu}=R_{\alpha\gamma\beta\mu}.$$ But $R_{\alpha\gamma\beta\mu}$ isn't one of the above listed symmetries. Where am I going wrong?


Some small questions about quantum spin and rotations

I'm studying about quantum-spin (in a syllabus about non-relativistic quantum- mechanics though), but I have some trouble understanding everything. So I would like to ask some small questions, which may perhaps clear up the picture as a whole. So they are trying to determine the influence of a spatial rotation, to the state in the two dimensional spin-space. So they try to determine the operator $Û_R$ responsible for this rotation (which acts on the spin-space and thus is represented by a 2x2 matrix). To do this they first try to determine $Û_{\epsilon}$ which is the infinitesimal unitary operator for rotation about an infinitesimal angle $\epsilon$ and they state that such an operator can always be written as: $$Û_{\epsilon}= 1-i\epsilonÂ$$ with $Â$ hermitic. So $Â$ is also acting on the two dimensional spin-space and can be represented by a 2x2 matrix. So far so good. But now they state that $Â$ has to be constructed in such a way that $<\chi\vert\chi>$ and $<\chi'\vert\chi'>$ and $<\chi'\vert\chi>$ are rotation- invariant. Where $\vert\chi>$ and $\vert\chi'>$ are states of the two dimensional spin-space and $\vert\chi'> = Û_{\epsilon}\vert\chi>$. What I don't understand is: 1) What is meant by rotation invariance? Because if we rotate $\vert\chi>$ and $\vert\chi'>$ by taking the action of a unitary rotation-operator $Û_{\theta}$ , then these inproducts are necessarily conserved, whatever $Â$ may be. So I guess they mean something else. 2) Why must $Â$ be constructed so that this is true? Perhaps this will be clear if I understand 1) though. Furthermore they state that this can only be the case if: $$Â = \vec{1}_n\cdot\vec{Â} = \frac{1}{2}\vec{1}_n\cdot\vec{\sigma}$$ With $\vec{Â}$ and $\vec{\sigma}$ hermitical vector operators. So I wonder: 3) Why is this only the case if this is true?   4) Somewhat further they also mention that $Tr(\sigma_z) = 0$ but I don't really see here too why this has to be the case. I hope that someone can answer one (or more) of these questions, I think it will help a lot in understanding the other claims of the chapter about quantum-spin.


Why doesn't the potential energy of any object equal 0

Consider a particle on the ground. This particle is raised by a force of magnitude $mg$ to a height $h$ above the ground. At this point, the work done on the particle by the force is $mgh$, which is equal to the potential energy of the particle. But, during this period, the force of gravity also acts on the particle and is displaced by $h$, and so does a work of $-mgh$ on the particle. Shouldn't the two cancel and no net work should be done on the particle? If they don't cancel, then where did the energy that came from the work done by the force of gravity go?


Do all massive bodies emit Hawking radiation?

It is known that any accelerated observer is subject to a heat bath due to Unruh radiation. The principle of equivalence suggests that any stationary observer on the surface of a massive body should also experience heat bath. Indeed by substituting surface gravity g into the formula for Unruh radiation one can find the black body temperature for a hypothetical hyper-cool planet: $$T = \frac{\hbar g}{2\pi c k}$$ which is $3.9766×10^{-20}\ K$ for Earth one even can find the time which it will take for Earth to evaporate: $5.69×10^{50}$ years. Since the heat in the super-cold Earth cannot come out of nothing one should assume that it will come from decay of particles due to a certain mechanism. Sometimes I heared an argument that an event horizon is needed for Hawking radiation to exist. But this can be countered by assumption of possibility of decay due to quantum virtual black holes (which inevitably should appear due to uncertainty principle, and the more massive and dense body is the greather concentration of virtual black holes inside it will be, eventually becoming similar to the concentration of bobbles inside a body of boiling water). Or just suggest that any massive body due to uncertainty principle can quantum tunnel into a black hole state so to emit Hawking radiation. So what is the conclusion here?   * Can we say that all massive bodies are surrounded by the atmosphere of heated vacuum? This is a weaker preposition: thermal state of surrounding vacuum does not mean energy transfer if the system is in thermodynamic equilibrium.   * Any body gradually evaporates, i.e. transfers its energy to the surrounding vacuum until completely vanishes? This is a stronger preposition and suggests emission of radiation al loss of mass.


Entropic force in polymers

According to my textbook, the elastic force in a rubber is caused to the tendency of the polymers to return to their initial disordered state of higher entropy. But isn't this looking at entropy on a larger level (I'm hesitant to say macroscopic) than it is usually looked at? Generally, we consider the entropy of individual atoms in an ordered structure (a crystal lattice, for instance). But here, we are looking at the entropy of entire polymer _chains_. Also, it just doesn't seem to make sense with the non-statistical view of entropy. The book explains it in the following way. The first law of thermodynamics, $$dE = dQ - dW$$ Making some substitutions, $$dE = TdS + Fdx$$ where $F$ is the elastic force, $dx$ is the displacement, $T$ is the temperature and $dS$ is the entropy change. Next, approximating $dE$ as zero, and solving for $F$, $$F = -T\frac{dS}{dx}$$ which is supposed to tell us the that the force is proportional to the rate $dS/dx$ at which the rubber band's entropy changes. But how is there any chage in entropy when no heat is being added? And, for that matter, if I take a string, lying all entangled, and try to straighten it, why doesn't _it_ exert an entropic force and try to go back to its higher entropic state?


Decompressing gas whilst driving a turbine - where does the energy come from?

Here's the situation: a friend is working on a civil engineering project to make some equipment to decompress liquefied natural gas before pumping to households. I don't know what the proper term for this bit of kit is calld but I will call it the **decompresser** Apparently what happens in these stations is that the gas is decompressed and because the gas cools down during expansion, it has to be heated back to something like room temperature before going on its way through the gas pipes to homes. They do this by burning kerosene. Now my friend says that they have designed a new decompresser that, in order to be more energy efficient, will generate electricity by decompressing the gas through a turbine. It is a long time since I did any related physics but the simpleton-scientist- thought-process that I have going on in my head is something like the following: -   * if the pressurised gas is having to do work turning the turbine, then this energy has to come from somewhere    * thus the temperature of the resulting de-pressurized gas will be lower.    * Because the temperature of the gas is lower, they would have to put more energy than they would have done to reheat the gas and they would not have any net energy gain. They would have electricity but they would be burning an equivalent amount of kerosene.  Now assuming that these civil engineers know a thing or two about thermodynamics, I readily admit that my thinking is wrong. I would very much appreciate it if someone could put me right on whether or not they can generate electricity as well as have the gas come out at the same temperature and pressure and could they not just have used the extra kerosene to generate the same amount of electricity (ignoring the various differing efficiencies of energy conversion that are going on). **PS.** Because I am clueless as to the actual answer, I will be relying heavily on the upvoting of + comments about the answers when it comes to marking an answer as the right 1!


Understanding bending light beam perpendicular to motion

I'm just reading a book about gravity. An example it gives is a spaceship accelerating. A beam of light travelling at right angles to the direction of movement of the spaceship enters it via a small pinhole. The book states that the beam of light would appear to bend to an observer within the spaceship. Intuitively this makes sense - if the ship were travelling at $0.5\,c$, for example, and was $10\,m$ across (from the pinhole to the opposing wall) then I'd expect the light to hit the wall $5\,m$ further back down (assuming the ship is accelerating "up") from the pinhole - the wall would have moved 5m further in the time it takes the light beam to cross the interior to the opposing wall I think I'm fundamentally misunderstanding though. The book states this is down to the acceleration of the ship and goes on to talk about how using equivalence the same bending would be a result of gravity. In my description above though, it's the velocity that causes the apparent bending, and the same would be visible if the ship was maintaining a constant $0.5\,c$. I suspect that if this were the case (constant $0.5\,c$) there would be no bending at all, but I'm not understanding why Can anyone enlighten me?


How much smaller will be human body, when we hypothetically get of every space between particles

I had an interesting dream, where advanced civilization compress their body with technology, that turn off space betweens atoms/particles. They travel in small spaceship with billions citizens very effectively. Crazy idea, but my question is how much smaller will the human body be, when we hypothetically get of every space between particles.


Could the Sun be partly powered by cosmic rays?

The Earth and the Sun has magnetic fields which shields us from cosmic rays, as a charged cosmic ray particle will loose kinetic energy when its direction is perpendicular to the magnetic field. So what happens to the kinetic energy of the cosmic ray particle? According to the first law of thermodynamics it can't just disappear. According to electro-mechanics it will induce an electric current. The particles also get a spiral motion and can induce a magnetic field like a current in a solenoid. Some radiation due to breaking of the particle (bremsstrahlung) may also occur. The vast magnetic field of our sun is bombarded with cosmic rays with far bigger energy intensity than the cosmic rays that hit our shielded Earth. On Earth the most energetic cosmic ray observed is a subatomic particle with kinetic energy equal to that of a baseball (5 ounces or 142 grams) traveling at about 100 kilometers per hour (60 mph). So I wonder if its a possibility that the Sun is partly powered by cosmic rays? How large can the total cosmic ray energy hitting the heliosphere be and where does the energy go?


Local tension and pressure measurements

We want to measure tensions and pressures on the surface of a tooth for an experiment in dental research. We want to do local measurements at different points on the tooth. The tooth is not a real one, but a model, so it would be possible to build a device into the tooth. The tensions and pressures we want to measure are of the order of a few kilo Pascal or potentially even less. Is there a reliable way to achieve such a measurement?


When we talk about speeds in relativity theory, where are they measured?

I recently asked a question here about if the direction we travel matters in relativity theory: Does it matter in which direction I travel in relativity theory? After I got answers and making more research about the topic, I found out that I was talking about speeds relative to the earth all the time. So, when I for example said that I travel away from earth at 50% of the speed of light, I automatically assumed that this is also the speed I can use to calculate the time dilation. But since the earth already travels at some always varying speed and thus is not standing still in space, I think I also need to put this into the calculation. I mean, if I would travel in the exact opposite direction the earth currently "flies" and at exactly the same speed, this would mean that I'm standing still and that the time for me would actually go as fast as possible, while the time on earth is still slowed down, as it always is due to its speed. So: What I wanted to show you is that it really seems to depend from where I actually measure the speed I travel and what I consider the "root". Moving away from earth at some speed can mean moving faster in space OR moving slower in space and thus also mean slower or faster time flow than the one on the earth. Am I right with this? Or am I totally wrong and it's valid to measure the speed I travel by how much the distance from me to the earth gets bigger? **UPDATE** Actually, I think I have understood the answer of @John Rennie, and it really seems to explain a lot of the issues I ran into before. His answer was, at least to my understanding, the following: If two objects in space stay together, they share the sime timeflow. If the distance between the objects changes over time, we cannot say which one will see the others clock going slower and which one will see the others clock going faster. To say that, we need to go back to the change of velocity that led to the increasing/decreasing distance. Here, we need to determine which of the objects changed it's velocity by the method John mentioned. And this will lead us to the answer. This also explains the twin paradox perfectly. But after doing some thought experiments I came across a scenario that I cannot explain: Imagine we humans have indeed built a death star as a moving base in space. This death star has also a lot of smaller fighter ships on board. Now the death star starts to fly away from earth at, say, 0.1c. According to the people on earth, the time on the death start now goes slightly slower than the time on the earth. After a while, a fighter on the death star begins to fly twice as fast (0.2c relative to the earth, 0.1c to the death star) in the same direction to explore the forthcoming space. According to the earthlings, the time on the fighter goes even slower than the time on the death star. The guys on the death star confirm that and notice that the time on the fighter seems to go slower compared to their time. After a while, the fighter found some enemy and decides to return to the death star to alert them. Here, the fighter has 2 options: 1) since the death star is also on its way, he could just stop and wait until the deathstar comes across. 2) Stopping, and moving into the other direction to inform the death star faster and before he runs into the enemies. The fighter goes for option 2) and flies back at 0.1c. For the earthlings, the fighter is now getting closer at 0.1c, while the death star is still moving away at 0.1c. For the death star, the fighter is getting closer at 0.2c. And that is the point where this starts getting weird: For the earthlings, the fighters speed decreased from 0.2c to 0.1c, which means that the fighters clock is now going faster, while still being slower than that on the earth. Actually, since the death star and the fighter are moving at the same speed relative to the earth, they share the sime timeflow. For the deathstar, the fighters speed increased from 0.1c to 0.2c, which means to them that the fighters clock is now going even slower than that on the death star. Can it really be?


Can the $\vec H$-field have non-zero divergence? Does an $\vec H$-field monopole exist?

We know that, $\vec \nabla \cdot \vec B=0$ but $\vec \nabla \cdot \vec H\neq 0$, if $\vec \nabla \cdot \vec M \neq 0$. Does it mean that, in those cases $\vec H$-field has poles although $\vec B$-field does not?


Does the Renormalization of QFT Contradict Canonical Quantization?

Does the renormalization of QFT contradict canonical quantization? In canonical quantization, you take the classical fields and canonical momenta and turn them into operators, and you require that the commutators ( or anti- commutators) equal the Poisson brackets of the original fields. Also, you need to quantize the classical Hamiltonian (of course, since classical fields commute but quantum operators don't, you need to quantize the Hamiltonian according to a well defined prescription, such as normal-ordering). The form of the quantum Hamiltonian will resemble the form of the classical one, but it will be an operator rather than a number. In particular, if a physical constant appears in the classical Hamiltonian, it will also appear in the Quantum one, and have the same value. Now, in renormalization the constants that appear in the (bare) Lagrangian are not the physical ones, and so are different from the constants that appear in the "classical" Lagrangian (and therefore Hamiltonian). So the "bare" quantum Hamiltonian will have a similar form to the classical Hamiltonian, but the constants will be totally (and perhaps infinitely) different! So does the quantization scheme need to be reformulated?


Modeling with First Order Equations

A ball with mass 015kg is thrown upward with initial velocity 20m/s from the roof of a building 30m high. There is air resistance of magnitude v^2/1325 directed opposite to the velocity , where the velocity is measured in m/s. a. Find the max height above the ground that the ball reaches. b. Find the time that the ball hits the ground. I am unsure how to set this one up.


Should any theory of physics respect the principle of conservation of angular momentum or linear momentum?

Is it possible that a theory that can describe the universe at the planck scale can violate things that we now consider fundamental in nature?For example can it violate rotational and translational invariance and subsequently momentum wouldn't be conserved ?Should one consider these invariance principles to be fundamental that we must choose the lagrangian to respect them ?


What forces act on a person sitting in a chair with wheels when they push off another stationary object

1. What is slowing me down when I push off my desk?   2. What are the "Major" contributors? and   3. Does how hard I push with one hand (or with two hands) make a significant difference?


Might a planet perform figure-8 orbits around two stars? I'm thinking that if the two stars were equal mass (and not orbiting each other) then a planet that were to go right between them would continue in a straight line, with no preference for either star. But since the two stars would in fact be orbiting each other, the system would be rotating and thus there would be a Coriolis preference for one of the stars. Might that preference be made to alternate stars? Another possibility would be if each star were in turn orbited closely by another planet, which would perform three orbits for each orbit of our planet of interest. Then things could be timed where on one pass star A's inner planet were aligned right to pull the planet of interest into an orbit, and on another pass star B's inner planet would be aligned right to pull the planet of interest into an orbit. So we have a system of five bodies, two massive (stars) each orbited by a minor, and one minor performing figure-8s. Is this at least plausible if contrived?


Double-slit expirement fundamentals (half-silvered mirror version)

In the double-slit experiment variation in which 2 half-silvered mirrors and 2 mirrors are used to illustrate the interference of a stream of photons or single photons at a given time step, how is it when the experiment is comprised of all 4 mirrors that the photon always knows to go to Camera 1/A in every illustration of the experiment I've ever heard of? Why doesn't it ALWAYS go to Camera 2/B instead with this setup? Seems so arbitrary and I have never come across the reason. Diagram to refresh you: (Assume all distances are equal in diagram.)                                         [CAM 1/A]                                   |        regular mirror             |             //--------------------\     [CAM 2/B]  (notice there is no beam to CAM 2/B)             |                     | half-silvered mirror #2             |                     |             |                     |             |                     |      [XXX>--\---------------------//  regular mirror             ^half-silvered mirror #1      [XXX> is the photon emitter/laser. The backslash to the right of the photon emitter represents one of the two half-silvered mirrors that I've designated as half-silvered mirror #1. So... why is it when interference occurs at half-silvered mirror #2 that the photon decides it prefers the camera designated CAM 1/A when it could easily just go to the other CAM every single time it interferes?


Diffraction of sound

The sound waves, by the virtue of it being a wave, shows diffraction and interference. But in diffraction, I learnt that if the wave is allowed to enter through a small aperture, there is a central maxima, along with several minima (zero intensity) on the sides. > **Why then in everyday life do we not encounter a situation in which we are > at one of those minima and fail to here any sound? For example,in a > soundproof studio room, suppose the narrow open door is the only entry point > for sound, can we then find a spot where we hear nothing going on outside > but by shifting to either side, we can hear it (although with different > intensities)?** Why aren't such phenomenon encountered in daily life, not being able to hear a source because of diffraction (or interference)? My question also relates to **radio waves** whose wavelengths is comparable to that of sound waves.


Supercharge definition in Becker-Becker-Schwarz

How does one interpret equation (4.20) in Becker-Becker-Schwarz "String Theory and M-Theory", particularly the $ \frac{\partial}{\partial \bar{\theta}^A} $ term where the $\theta$ is the Grassmann coordinates of the super-world-sheet {$\sigma^\alpha,\theta_A$} and A is the spinor index {+,-}? The link to the text is: http://www.cambridge.org/us/academic/subjects/physics/theoretical-physics-and- mathematical-physics/string-theory-and-m-theory-modern-introduction?format=HB


How to explain in simple terms why Entanglement is more than just complicated hidden variables

I haven't taken a graduate level physics course on quantum mechanics so I get lost in the strange looking equasions. It's hard for me too see in any of the explanations of how quantum computers and entanglement work without thinking that it looks like a big parlor trick. It's like a magician taking a black and white marble shuffling them in a cup then taking one in each hand without anyone seeing which is in which hand and then revealing a white marble in his left hand and declaring: >   1. I can predict with absolute certainty that the marble in my right hand > is black. >   2. Until I had actually observed the white marble in my left hand, neither > marble actually had a definite color. They were just a superposition of two > possibilities. > A magic trick isn't very convincing unless the audience can understand it and not explain in simpler terms. Are there experimental results that the average person can understand and not explain using classical intuition?


Big Bang and Spherical Symmetry

If the universe did indeed start with the big bang why is the universe not spherically symmetric? As per Wikipedia entry on Big Bang, (and my understanding as well) big bang is the best explanation as of now. > The Big Bang is the scientific theory that is most consistent with > observations of the past and present states of the universe, and it is > widely accepted within the scientific community. Basically considering the universe as a sphere (all points equidistant from a center) for any given state at any given point, there should have existed an identical state at the other end (line passing through the center) of the sphere. Also, all points equidistant from the center should have had the same 'state'. What is the explanation for the 'asymmetry' that we have today?


Anticommutatorrelation in Bogoliubov-de Gennes Hamiltonian

I almost solved the problem Equivalence of Bogoliubov-de Gennes Hamiltonian for nanowire. In the next steps I used the notation by arXiv:0707.1692: $$ \Psi^{\dagger} = \left(\left(\psi_{\uparrow}^{\dagger}, \psi_{\downarrow}^{\dagger}\right), \left(\psi_{\downarrow}, -\psi_{\uparrow}\right)\right) $$ and $$ \Psi = \left(\left(\psi_{\uparrow}, \psi_{\downarrow}\right), \left(\psi_{\downarrow}^{\dagger}, -\psi_{\uparrow}^{\dagger}\right)\right)^{T}\text{.} $$ I'm trying to show that the Hamiltonian for a nanowire with proximity-induced superconductivity $$ \hat{H} = \int dx \text{ } \left[\sum_{\sigma\epsilon\\{\uparrow,\downarrow\\}}\psi_{\sigma}^{\dagger}\left(\xi_{p} + \alpha p\sigma_{y} + B\sigma_{z}\right)\psi_{\sigma} + \Delta\left(\psi_{\downarrow}^{\dagger}\psi_{\uparrow}^{\dagger} + \psi_{\uparrow}\psi_{\downarrow}\right)\right]\text{,} $$ can be written as $$ \hat{H} = \frac{1}{2}\int dx \text{ } \Psi^{\dagger}\mathcal{H}\Psi $$ with $\mathcal{H} = \xi_{p} 1\otimes \tau_{z} + \alpha p \sigma_{y}\otimes\tau_{z} + B\sigma_{z}\otimes 1 + \Delta 1\otimes\tau_{x}$ (here $\tau_{i}$ are the Pauli matrix for the particle-hole space and $\otimes$ means the Kronecker product). Here I calculate as example the first und third term of $\Psi^{\dagger}\mathcal{H}\Psi$. $$ \tau_{z}\Psi = \left(\left(\psi_{\uparrow}, \psi_{\downarrow}\right), -\left(\psi_{\downarrow}^{\dagger}, -\psi_{\uparrow}^{\dagger}\right)\right)^{T} = \left(\left(\psi_{\uparrow}, \psi_{\downarrow}\right), \left(-\psi_{\downarrow}^{\dagger}, \psi_{\uparrow}^{\dagger}\right)\right)^{T} $$ $$ \Rightarrow \left(\left(\psi_{\uparrow}^{\dagger}, \psi_{\downarrow}^{\dagger}\right), \left(\psi_{\downarrow}, -\psi_{\uparrow}\right)\right)\xi_{p}\left(\left(\psi_{\uparrow}, \psi_{\downarrow}\right), \left(-\psi_{\downarrow}^{\dagger}, \psi_{\uparrow}^{\dagger}\right)\right)^{T} = \left(\psi_{\uparrow}^{\dagger}, \psi_{\downarrow}^{\dagger}\right)\xi_{p}\left(\psi_{\uparrow}, \psi_{\downarrow}\right)^{T} + \left(\psi_{\downarrow}, -\psi_{\uparrow}\right)\xi_{p}\left(-\psi_{\downarrow}^{\dagger}, \psi_{\uparrow}^{\dagger}\right)^{T} = \psi_{\uparrow}^{\dagger}\xi_{p}\psi_{\uparrow} + \psi_{\downarrow}^{\dagger}\xi_{p}\psi_{\downarrow} - \psi_{\downarrow}\xi_{p}\psi_{\downarrow}^{\dagger} -\psi_{\uparrow}\xi_{p}\psi_{\uparrow}^{\dagger} $$ Now I use the anticommutatorrelation $\\{\psi_{\sigma}, \psi^{\dagger}_{\sigma^{\prime}}\\} = \delta_{\sigma,\sigma^{\prime}} \Leftrightarrow \psi_{\sigma}\psi^{\dagger}_{\sigma} = 1 - \psi_{\sigma}^{\dagger}\psi_{\sigma}$ $$ \Leftrightarrow 2\psi_{\uparrow}^{\dagger}\xi_{p}\psi_{\uparrow} + 2\psi_{\downarrow}^{\dagger}\xi_{p}\psi_{\downarrow} - 2\xi_{p} $$ However, the term $-2\xi_{p}$ here are wrong. For the third term I obtain $$ \psi_{\uparrow}^{\dagger}B\sigma_{z}\psi_{\uparrow} + \psi_{\downarrow}^{\dagger}B\sigma_{z}\psi_{\downarrow} + \psi_{\downarrow}B\sigma_{z}\psi_{\downarrow}^{\dagger} +\psi_{\uparrow}B\sigma_{z}\psi_{\uparrow}^{\dagger} = \psi_{\uparrow}^{\dagger}B\sigma_{z}\psi_{\uparrow} + \psi_{\downarrow}^{\dagger}B\sigma_{z}\psi_{\downarrow} - \psi_{\downarrow}^{\dagger}B\sigma_{z}\psi_{\downarrow} -\psi_{\uparrow}^{\dagger}B\sigma_{z}\psi_{\uparrow} + 2B\sigma_{z} = 2B\sigma_{z} $$ Does anybody see my mistake?


Understanding wave functions

I'm currently writing an essay on the measurement problem, and I'm not quite certain that I've fully understood the purpose of the wave function, in that does the following sentence make sense with respect to the wave function: > _Although the wave function itself is never physically observed, we are > given resultant macroscopic events which we can interpret, allowing us to > formulate a conclusion._ What I mean to say is that although we don't actually see the wave function, we can see the kind of after-math of what happens in the form of a physical macroscopic event..is that right?


Casimir Effect and polarization of photons

I have read Casimir's derivation of the Casimir fore between 2 parallel plates and have been told that in reality, the Casimir force should be twice as large due to the 2 polarization states of photons. I don't understand how/why the 2 polarization states affects the force. Could anyone kindly explain?


Action Reaction when I pushing a trolley?

I tried to explain how those force work but I can hardly figure it out. I exerted a force on the trolley and there will be a force on trolley on me as well. This is the newton's third law. But why the trolley will move? Where's the force come from


Has anyone studied a statistical scaling law for the universe?

How do named objects in the universe scale? Is there a predictable curve for an ordered list, say {atom, animal, planet, solar system, galaxy, etc}? Can you then use the analysis to predict when the next larger structure will be seen?


Sign Conventions for Dirac equation

Is it possible to have the Dirac sign convention, (-,+,+,+) and at the same time use the metric $$dt^2-dx^2-dy^2-dz^2$$ i.e have opposing Dirac and metric tensor conventions?


Rotation motion like the number 8

I'm a college student majoring in culinology and I'm trying to find out the reason or method of the number 8 motion. Responses doesn't have to be in culinology examples, but that would be a great help if it is. From my observations of the circular motions v.s. the figure 8 motions, if I was to mix cake batter with big circular motions in a bowl; the center would not be affected. Mixing the cake batter with small circular motions would have the opposite effect as making big circles. Now mixing the cake batter in a figure 8 motion would hit all the spots, (center & outer area) but you'll have to turn the bowl to make this true. I think the amount of velocity and the force used to mix in these ways would change things.


Electromagnetic fields vs electromagnetic radiation

As I understand, light is what is more generally called "electromagnetic radiation", right?   The energy radiated by a star, by an antenna, by a light bulb, by your cell phone, etc.. are all the same king of energy: electromagnetic energy, i.e. photons traveling through space. So far, so good? (if not please clarify) On the other hand, there are these things called "electromagnetic fields", for example earth's magnetic field, or the magnetic field around magnets, the electric field around a charge or those fields that coils and capacitors produce. Now here is my question:   * Are these two things (electromagnetic fields and electromagnetic radiation) two forms of the same thing? or they are two completely different phenomena?    * If they are different things, What does light (radiation) have to do with electromagtetism? I'm not asking a complex theoretical answer, just an intuitive explanation. Thanks.


Normal force at the inclined plane

A particle mass $m$ moves down a frictionless inclined plane. Does a normal force cancels a normal component of $m$ **g**?


Effective Field Theories of QCD

Recently, I am studying the online course Effective Field Theory provided by MIT OCW. Prof. Stewart gives a nice picture to summarize the effective theories: ![Example Effective Field Theories derived from Quantum Field Theory](http://i.stack.imgur.com/lXZ23.jpg) As a newbie in this field (I only have a knowledge of QFT on the level of Peskin & Schroeder's book), I am quite confused with so many terminologies. What do they stand for and Why do we need these different theories? Could someone explain this picture in depth (especially on their advantages and limitations)? Any help will be greatly appreciated.


Meaning of $k$ in Sachs-Wolfe formula for angular power spectrum

I've seen the formula for the angular power spectrum of the CMB written as $$C_\ell = \frac2\pi \int\left|\Theta_\ell(k) \right|^2 k^2dk, $$ where $\Theta_\ell(k)$ is the temperature contrast at a given wavevector $k$ (the specific reference I'm using is the book "Primordial Cosmology" by Peter and Uzan (2009), pg. 313). The expression for $\Theta_\ell(k)$ is derived from linear perturbation theory in a cosmological background, using _conformal time_ $\eta$, as opposed to cosmic time $t$. So it seems to me that $k$ in this formula is _not_ the physical wavenumber, but rather the conformal wavenumber: $$ k = ak_\text{phys},$$ where $a$ is the scale factor. My issue is that if I'm using the conformal wavevector $k$ in the above formula for $C_\ell$, it seems to be non-gauge invariant. I can make a transformation where I send $a\rightarrow a/\lambda$, $k\rightarrow k/\lambda$, $\eta\rightarrow \lambda \eta$, and $x^i\rightarrow \lambda x^i$. This transformation doesn't seem to affect $\Theta_\ell(k)$, but does rescale $k$ so that the transformation of $C_\ell$ is $$C_\ell\rightarrow C_\ell/\lambda^3 $$ This does not seem like a very good feature for a physical observable. My question is how do I get a scaling invariant observable, and more generally what is the correct interpretation of $k$ in that formula?


Calculating Dynamics of a Thick Sheet of Material

So I'm working on a computational neuroscience problem, and I'm attempting to model skin dynamics for use in some perception models. The problem is as follows: Imagine you had a two dimensional sheet with some thickness. Imagine stimulating it with a probe with negligible spatial extent (i.e. a point source of stimulation) which can deliver arbitrary time varying stimuli (let's say for our purposes it's a sinusoid). The question is basically how can you determine $z(x,y,t)$, the lateral position of the sheet as a function of time for each point on the sheet, given knowledge of $z(0,0,t)$ (which is just how the probe is stimulating the sheet). The idea is basically that the probe will induce travelling waves that will propagate to every other point on the sheet. Now, imagine instead that your probe has spatial extent, in other words, now you know $z(x,y,t)$ for $(x,y)\in A\subseteq \mathbf{R}^2$ where $A$ is essentially the shape of the probe (which we can assume to be simple for now, like a circle or square). Then how can we determine $z(x,y,t)$ for $(x,y)\in\mathbf{R}^2$? My suspicion is that you could use the solution to the problem above and essentially integrate over $A$ (i.e. compute the response of the sheet elsewhere as the summation of all the contributions of the point sources of $A$). Finally, imagine instead of a sheet, you had some non-convex shape (like the hand for instance). How would you compute the sheet (skin) dynamics at another point given knowledge of how the probe is moving laterally. I think we've solved the first problem. Essentially, we experimentally derive propagation speeds and how sinusoids decay as a function of distance from the point at which the probe contacts the skin (for instance, we may decide that if you're stimulating the skin with a wave $z(0,t) = A\cos(2\pi f t)$, then $$z(r,t) = \frac{1}{r^{\gamma(f)}}z(0,t)$$ where the decay parameter as a function of frequency is experimentally derived. Then we extend this to arbitrary stimuli by decaying the wave in fourier space and then inverting it back into a time series. I'm wondering $(a)$ Is this solution correct even in the simple case? $(b)$ How does the solution change when you incorporate the spatial extent of your probe? $(c)$ How do you compute the skin dynamics at another point, where there isn't a straight line connecting contact point of the probe to the point in question? Any help would be greatly appreciated, or at least reference to some literature that might help me begin to think about how to solve this problem?