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Why does the sound pitch increase on every consecutive tick at the bottom of a filled cup of coffee? Since I don't know the proper physical terms for this, I describe it in everyday English. The following has kept me wondering for quite some time and so far I haven't found a reasonable explanation.
When you fill a ceramic cup with coffee and you click with the spoon at the bottom (from the top, through the coffee), each following tick, even when you pause for some seconds, will have a higher pitch. The following I've observed so far:
*
*works better with coffee than with tea (works hardly at all with tea)
*works better with cappuccino than with normal coffee
*doesn't work with just cold water
*works best with ceramic cups, but some plastic cups seem to have the same, yet weaker, behavior
*doesn't work on all types of cups, taller cups seem to work better
*must have a substantive amount of liquid (just a drop doesn't make it sing).
It must be something with the type of fluid, or the milk. I just poured water in a cup that had only a little bit fluffy left from a previous cappuccino, and it still worked. Then I cleaned it and filled it again with tap water and now it didn't work anymore.
Can someone explain this behavior?
| I first noticed this in a hot cup of Horlicks, made with milk. I would stir in the powder vigorously, then tap the bottom of the cup with the spoon to check that all the powder had dissolved. Even two taps, one second apart is enough to detect the rising pitch. It continues rising and rising over the course of, perhaps, 20 seconds.
The interesting thing is that you can make the pitch drop again by stirring it up again. It seems that the pitch is directly related to the rate at which the milk is spinning.
| {
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Are we inside a black hole? I was surprised to only recently notice that
An object of any density can be large enough to fall within its own
Schwarzschild radius.
Of course! It turns out that supermassive black holes at galactic centers can have an average density of less than water's. Somehow I always operated under the assumption that black holes of any size had to be superdense objects by everyday standards. Compare the Earth to collapsing into a mere 9mm marble retaining the same mass, in order for the escape velocity at the surface to finally reach that of light. Or Mt. Everest packed into one nanometer.
Reading on about this gravitational radius, it increases proportionally with total mass.
Assuming matter is accumulated at a steady density into a spherical volume, the volume's radius will only "grow" at a cube root of the total volume and be quickly outpaced by its own gravitational radius.
Question:
For an object the mass of the observable universe, what would have to be its diameter for it to qualify as a black hole (from an external point of view)?
Would this not imply by definition that:
*
*The Earth, Solar system and Milky Way are conceivably inside this black hole?
*Black holes can be nested/be contained within larger ones?
*Whether something is a black hole or not is actually a matter of perspective/where the observer is, inside or outside?
| The entropy of a black hole is maximized. This is not the case for the matter that makes up the Earth; the entropy is not zero, but it's not maximized either. Thus, we do not live in a black hole.
| {
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Is chaos theory essential in practical applications yet? Do you know cases where chaos theory is actually applied to successfully predict essential results? Maybe some live identification of chaotic regimes, which causes new treatment of situations.
I'd like to consider this from the engineers point of view. By this I mean results that are really essential and not just "interesting". For example one might say weather calculation effects are explained with chaos theory, but an "engineer" might say: "So what? I could have told you without a fancy theory and it provides no added value since there is nothing I can change now."
| I think it depends on the meaning of "application". In the wikipedia entry there are a lot of applications of chaos theory listed:
Chaos theory is applied in many scientific disciplines, including: geology, mathematics, microbiology, biology, computer science, economics, engineering, finance, meteorology, philosophy, physics, politics, population dynamics, psychology, and robotics.
For example, in engineering there is the following in the abstract of a paper:
Control of chaos: Methods and applications in engineering ☆A survey of the emerging field termed “control of chaos” is given.
Several major branches of research are discussed in detail: feedforward or “nonfeedback control” (based on periodic excitation of the system); “OGY method” (based on linearization of the Poincaré map), “Pyragas method” (based on a time-delay feedback), traditional control engineering methods including linear, nonlinear and adaptive control, neural networks and fuzzy control. Some unsolved problems concerning the justification of chaos control methods are presented. Other directions of active research such as chaotic mixing, chaotization, etc. are outlined. Applications in various fields of engineering are discussed.
Chaos theory is not chaotic in the everyday sense, there are those strange attractors and the behavior of collective solutions to dynamical equations, which is what chaos theory deals with, may very well give a handle to control complex systems. It seems to still be at the exploration and research phase, but yes, there already seem to be possible applications.
| {
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What are some ways that humans could have influence over what sequence a star was in? How would a society go about either preventing our sun in its primary sequence from going into a Red Giant a billion years from now? Or perhaps, accelerating the process of going from main sequence of our start to a red giant prematurely.
What catalysts would have to happen, implemented by humans, so no natural phenomenon could play a part in it, that would have an impact, even slight, on the future processes of the sun's lifecycle.
Any thoughts or help would be great!!!!
| Sci-fi method: It might still be a competitive in terms of energy expenses options to put all the people into a spacecraft, attain ultrarelativistic velocities, wait a second (or some other necessary time), and then come back to what is left from the Earth. Due to special relativistic time dilation effects during the flight billions of years may pass for the Sun to end its main-sequence phase.
| {
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Why does a glass rod when rubbed with silk cloth aquire positive charge and not negative charge? I have read many times in the topic of induction that a glass rod when rubbed against a silk cloth acquires a positive charge. Why does it acquire positive charge only, and not negative charge?
It is also said that glass rod attracts the small uncharged paper pieces when it is becomes positively charged. I understand that a positively charged glass rod attracts the uncharged pieces of paper because some of the electrons present in the paper accumulate at the end near the rod, but can't we extend the same argument on attraction of negatively charged silk rod and the pieces of paper due to accumulation of positive charge near the end?
| This is because glass is above silk in the triboelectric series (attracts electrons less than silk) and when rubbed, silk 'takes' its electrons.
And yes, if you had a silk rod it would also attract neutral paper, because paper pieces are turned into dipoles, as you explained.
| {
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How does an object regain its neutrality after being charged by rubbing? Objects (like combs) can be charged by rubbing as charged particles, particularly electrons, are transferred from one object to other. This can be seen as the object (comb) attracts small bits of paper. After some time, the charge on the body seems to disappear. How does this charge disappear without any external influence?
| Static Electricity
Static electricity is the opening of the fabric of space, or of very tiny black holes in the fabric of space time.
The attractiveness of matter or objects are being pulled into these little opening of fabric of space until the holes close as the fabric of space regain it's ground or natural state.
Randy Lee Holmes 2003
| {
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How to derive addition of velocities without the Lorentz transformation? Lorentz contraction and time dilatation can be deduced without Lorentz transformation. Can you deduce also the theorem of addition of velocities
$$w~=~\dfrac{u+v}{1+uv/c^2}$$
without Lorentz transformation? Using just the constancy of light speed.
| I endorse Ron's answer – it's the systematic way to proceed. The velocity $v/c$ may be written as $\tanh \eta$ where $\eta$, the rapidity or whatever, is the hyperbolic (Minkowski) counterpart of the (Euclidean) angle. The addition of velocities then boils down to an addition formula for $\tanh(\eta_1+\eta_2)$ because the rapidities just add additively.
Let me offer an elementary derivation without any fancy rapidities. Imagine that an object moves by the speed $u$ to the right, another object moves by $v$ to the left with respect to our frame. What is their relative speed?
The world line of the first observer is a straight line containing the points $(0,0)$ and $(1,u)$; the coordinates are $(t,x)$. The other object has a world line connecting $(0,0)$ with $(1,-v)$. Now, let's imagine that we tranform the situation into the rest frame of the second observer, i.e. boost it by the velocity $v$. How will the first observer's world line tilt?
To find the answer, note that by making the Lorentz boost which fixes the origin $(0,0)$, the Lorentzian inner product of the two vectors, $(1,u)$ and $(1,-v)$, will not change; I define the inner product of $(A,B)$ and $(C,D)$ as $AC-BD/c^2$ where the minus sign comes from the Lorentzian relativity twists and $c^2$ is the conventional conversion of length to time. Their length won't change, either. It also means that the inner product divided by the product of lengths won't change. In the original frame, it is equal to
$$ \frac{(1,u)\cdot (1,-v)}{|(1,u)|\cdot |(1,-v)|} = \frac{1^2+uv/c^2}{\sqrt{1-u^2/c^2}\sqrt{1-v^2/c^2}} $$
Note that by taking the ratio, I canceled the absolute normalization of the two vectors $(1,u)$ and $(1,-v)$, so this normalization doesn't matter. However, this ratio must be the same in the new frame where the observers' vectors indicating the directions of the world line are $(1,0)$ and $(1,V)$ where $V$ is the total relative velocity. From those two vectors, the same ratio as above (which again cancels the normalization) is equal to
$$ \frac{(1,0)\cdot (1,V)}{|(1,0)| \cdot |(1,V)|} = \frac{1}{\sqrt{1-V^2/c^2}} $$
Just to be sure, the ratios have to be equal and they may also be written as $\cosh \eta$ where $\eta$ is the total "hyperbolic angle" i.e. rapidity in between the two world lines, the same "angle" as discussed above. The formula for $\cosh$ is analogous to the high school formula for $\cos$ involving the inner product but you don't need to know anything from this paragraph to follow my derivation.
Now, we have
$$ \frac{1}{\sqrt{1-V^2/c^2}} = \frac{1^2+uv/c^2}{\sqrt{1-u^2/c^2}\sqrt{1-v^2/c^2}}$$
Square it and invert it:
$$ 1 - \frac{V^2}{c^2} = \frac{(1-u^2/c^2)(1-v^2/c^2)}{(1+uv/c^2)^2} $$
Expand the product in the numerator and subtract one from both sides:
$$ - \frac{V^2}{c^2} = \frac{1 - u^2/c^2 - v^2/c^2 + u^2 v^2/c^4 - 1 - 2uv/c^2-u^2 v^2/c^4}{(1+uv/c^2)^2} $$
The numerator of the right hand side simplifies, two pairs of terms cancel:
$$ - \frac{V^2}{c^2} = \frac{- u^2/c^2 - v^2/c^2 - 2uv/c^2}{(1+uv/c^2)^2} $$
Now, multiply both sides by $(-1)$ to get rid of the signs. And even I am able to compute the square root now:
$$ \frac{V}{c} = \frac{u/c+v/c}{1+uv/c^2} $$
which we wanted to prove. Feel free to multiply it by $c$ again.
| {
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Can relativistic kinetic energy be derived from Newtonian kinetic energy? Relativistic kinetic energy is usually derived by assuming a scalar quantity is conserved in an elastic collision thought experiment, and deriving the expression for this quantity. To me, it looks bodged because it assumes this conserved quantitiy exists in the first place, whereas I'd like a derivation based upon using KE $= \frac12 mv^2$ in one frame, and then summing it in another frame say to get the total kinetic energy. Can this or a similar prodecure be done to get the relativistic kinetic energy?
| Assuming energy conservation isn't "bodged" because at the most fundamental level, energy is defined as the quantity that is conserved as the result of the time-translational symmetry. All specific formulae for energy, such as $mv^2/2$ in nonrelativistic mechanics, are just solutions to the problem "find a conserved quantity linked to that symmetry".
Still, you can try to achieve what you have defined. First, you must realize that $K=mv^2/2$ only holds if $v\ll c$: it's just not a valid formula in relativity for large velocities. It seems that you believe that $E=mv^2/2$ is correct in some frames even in relativity but it's not. Your formula is just an approximation, via Taylor expansions,
$$ \frac{mc^2}{\sqrt{1-v^2/c^2}} = mc^2 + \frac{mv^2}{2} + \frac{3mv^4}{8c^2} + \dots $$
If you want to use $E=mv^2$ for small $v$ and deduce what is $E$ for an arbitrary $v$ comparable to the speed of light $c$, you must use infinitely many interpolating inertial systems.
In this SE question
How to derive addition of velocities without the Lorentz transformation?
Ron Maimon explained how velocities add. So if you want to switch to an inertial system moving by velocity $v$, you may calculate a rapidity from
$$\tanh a = \frac vc$$
These rapidities behave as angles so if you're boosting by some incremental speeds many times, the rapidities just add up (much like angles for rotations). The total energy is then $mc^2\cdot \cosh a$ which is equal to the usual relativistic formula but the derivation of this fact will have to use some conservation of energy argument similar to one you know. The relativistic formula is the only one that reduces to $mv^2/2$ for infinitesimal $v$ and that conserves the energy while the object is boosted.
| {
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What's wrong with this equation for harmonic oscillation? The question:
A particle moving along the x axis in simple harmonic motion starts
from its equilibrium position, the origin, at t = 0 and moves to the
right. The amplitude of its motion is 1.70 cm, and the frequency is
1.10 Hz. Find an expression for the position of the particle as a function of time. (Use the following as necessary: t, and π.)
Using the equations:
$$
x(t) = A \cos(\omega t + \Phi)
$$
$$
\omega = 2\pi f
$$
I get A = 1.7cm or 0.017m, and
$$
\omega = 6.91
$$
I know that t = 0, x = 0. Thus,
$$
0 = 0.017 \cos(\Phi )
$$
And therefore,
$$
\Phi = \pi / 2
$$
From all of this, it seems to me that the equation for position with respect to time should be:
$$
x = 0.017 \cos(6.91t + \pi/2)
$$
Am I doing something wrong, because the above is not getting checked as the right answer (it's an online homework)
| The cosine has more than one zero. And the text specifies that the particle goes to the right (I assume that the x axis also goes to the right). Now in which direction does the cosine go at $\pi/2$? And where's another zero?
| {
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Why there is a $180^{\circ}$ phase shift for a transverse wave and no phase shift for a longitudinal waves upon reflection from a rigid wall? Why is it that when a transverse wave is reflected from a 'rigid' surface, it undergoes a phase change of $\pi$ radians, whereas when a longitudinal wave is reflected from a rigid surface, it does not show any change of phase? For example, if a wave pulse in the form of a crest is sent down a stretched string whose other end is attached to a wall, it gets reflected as a trough. But if a wave pulse is sent down an air column closed at one end, a compression returns as a compression and a rarefaction returns as a rarefaction.
Update: I have an explanation (provided by Pygmalion) for what happens at the molecular level during reflection of a sound wave from a rigid boundary. The particles at the boundary are unable to vibrate. Thus a reflected wave is generated which interferes with the oncoming wave to produce zero displacement at the rigid boundary. I think this is true for transverse waves as well. Thus in both cases, there is a phase change of $\pi$ in the displacement of the particle reflected at the boundary. But I still don’t understand why there is no change of phase in the pressure variation. Can anyone explain this properly?
| Here is another possible way of explanation:
Reflection of the wave is similar process as crushing two waves, one from the left and one from the right, which meet exactly at the surface. Now, if you wish that particle at the surface has zero displacement, then the wave on the right must be point-symmetrical through that particle to the wave on the left.
If left wave pulls particle at the surface up, the right wave must pull it down. If left wave pulls particle at the surface down, the right wave must pull it up.
Obviously, trough and crest match.
If left wave pulls particle at the surface left, the right wave must pull it right.
However, pulling left from the left and right from the right both corresponds to rarefaction.
If left wave pushes particle at the surface right, the right wave must push it left.
However, pushing right from the left and pushing left from the right both corresponds to compression.
I really love this problem (as mentioned above) but this explanation is furthest my mind is able to go...
| {
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Equations instead of psychrometric charts I want to create a program that will accurately simulate a condensor. I want to use the data in psychrometric charts. But I cannot and hence want to use equations that show similar data. Any idea where to start?
| There are existing software libraries of psychrometric functions available. These could be suitable for use directly, or at least as points of reference if for some reason you need to create your own. i.e., you can inspect the formulae, coefficient values, etc.
One set of psych equations is the PsychroLib project:
https://github.com/psychrometrics/psychrolib
PsychroLib is a library of functions to enable the calculation of
psychrometric properties of moist and dry air. Versions in Python, C,
Fortran, JavaScript, and Microsoft Excel Visual Basic for Applications
(VBA) are available. The library works in both metric (SI) and
imperial (IP) systems of units. For a general overview and a list of
currently available functions, please see the overview page.
A second source would be EnergyPlus which uses psychrometric functions for various building energy analysis purposes.
EnergyPlus has a full complement of psychrometric functions. All the
routines are Fortran functions returning a single precision real
value.
*
*Documentation
*Code
The code actually appears to have decent comments in addition to the actual documentation.
| {
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Why is glass a good conductor of heat? AFAIK Glass is insulator, it doesn't have free electron. It's said metal is a good conductor of heat because it has free electron, glass doesn't have free electron, why it is a good conductor of heat?
| There are at least two mechanism of thermal conductivity - free electrons and thermal phonons. The first mechanism can be prevalent in metals, the second one is important in dielectrics. I did not look up thermal conductivity of glass, but such excellent dielectric as diamond has higher thermal conductivity than any metal, as far as I know.
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Is the Avogadro's constant equal to one? Question: Is the Avogadro's constant equal to one?
I was tasked with creating a presentation on Avogadro's work, and this is the first time I actually got introduced to the mole and to Avogadro's constant. And, to be honest, it doesn't make any mathematical sense to me.
1 mole = 6.022 * 10^23
Avogadro's constant = 6.022 * 10^23 * mole^(-1)
What?
This hole field seems very redundant. There are four names for the same thing! Since when is a number considered to be a measurement unit anyway?!
| Yes it's a little odd to have a unit of 'amount'. At least in English, it might make more sense in other languages
The second line is really "Avogadro's constant = 6.022 * 10^23 * items/mole"
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Operators Uncertainty $\hat A$ is an operator.
The uncertainty on $\hat{A}$, $\Delta A$ is defined by:
$$\Delta A=\sqrt{\langle\hat A^2\rangle - \langle\hat A\rangle^2}$$
what is difference between
$\langle\hat A^2\rangle$ and $\langle\hat A\rangle^2$
that leads to Uncertainty Relation between two Operators?
more details:
$$
\langle\hat A^2 \rangle=\langle\psi|\hat A^2|\psi \rangle$$
What is the name of difference between absolute value of these two complex conjugates
| Although Qmechanics's answer is formally complete and correct, there is a more intuitive formulation of this identity that makes it self evident. Consider the operator B which is A minus its expectation value in some state.
$$B = A - \langle A\rangle $$
Then the expectation value of B is zero in the same state (obviously--- it has been shifted to make it so). The expected value of $B^2$ can be nonzero--- it is a measure of the spread in B in state $\psi$. It is positive, as you can see by the definition of matrix multiplication (or by "inserting the identity in a basis")
$$ \langle B^2 \rangle = \sum_i \langle |B|i\rangle\langle i|B\rangle $$
The last thing on the right is the sum of positive quatities of the form $c^*c$. If you now reexpress the expectation value of $B^2$ in terms of A,
$$ \langle B^2 \rangle = \langle (A-\langle A\rangle)^2\rangle = \langle A^2\rangle - 2 \langle A\langle A\rangle \rangle + \langle A\rangle^2 = \langle A^2\rangle - \langle A\rangle^2 $$
This manipulation justifies this thing.
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Proton-Neutron Lattice as a form of matter? Would it be possible for a lattice of protons and neutrons (I'm picturing a plane of hexagons in my head) to exist bound by the strong nuclear force (not gravity)? I know that the strong force losses its power when an atomic nucleus gets to be too large, but in a lattice, it would only have to bond one proton to a few neutrons or one neutron to a few protons at a time. Would this work? If so, what would be the properties of such a material?
| No, because the repulsive power of the protons accumulates toward infinity, blowing the structure apart. To accomplish what you have described you would need stable, negatively charged particles like antiprotons... which of course also fail in such an arrangement because they annihilate the protons!
The instability from accumulating positive charge is also why higher elements become unstable.
You can create crystalline structures that are composed mostly of neutrons with electron-neutralized protons sprinkled in, but only if you have very powerful compression to keep the neutrons stable and the whole arrangement bound together. That's the material in neutrons stars, sometimes called "neutronium" (yes, as in Star Trek), and it's very, very stiff. Breaks in the neutron crusts of such stars lead to abrupt changes in the otherwise incredibly precise timing of pulsar radio bursts.
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The Difference between Thomas-Fermi Screening and Lindhard Screening Assuming the general theory of screening related to electron-electron interactions, I was wondering if anyone could provide a clear, yet conceptually complete explanation of the differences between the Thomas-Fermi and the Lindhard theories?
Following the treatment by Ashcroft and Mermin (1976),I get the impression that the main difference is that the Thomas-Fermi model assumes that the electrostatic potential is slowly-varying in $\vec{r}$: $$E_i(\vec{k}) = \frac{\hbar^2k^2}{2m}-e\phi(\vec{r})$$, but I'm having trouble grasping the full physical significance of this statement.
| First, the Thomas-Fermi screening is a semiclassical static theory which assumes that the total potential $\phi(\mathbf{r})$ varies slowly in the scale of the Fermi length $l_{\text{F}}$, the chemical potential $\mu$ is constant and that $T$ is low. In principle, it does not rely on linear response theory.
The condition of slowly varying potential is a general condition of validity of semiclassical models. Physically, if the particle [electron] is represented by a wave packet, what is tellying us is that all the waves in the wavepacket will see the same potential and the particle will suffer [or enjoy!] a force as if it was point-like ["classical"] because such potentials gives rise to ordinary forces in the equation of motion describing the evolution of the position and wavevector of the packet. The wavepacket must have a well-defined wavevector on scale of the Brillouin zone [thus $\Delta k \simeq k_{\text{F}}$] and therefore can be spread in the real space over many primitive cells.
Mathematically, the assumption that your potential is a slowly varying function of the position implies that the theory is not valid for $|\mathbf{q}| \gg k_{\text{F}}$ [and therefore for $|\mathbf{r}| \ll l_{\text{F}}$].
On the other hand, the static Lindhard dielectric function is a fully quantum treatment of the problem and it is valid for all the ranges of $\mathbf{q}$. It includes, in the limit $\mathbf{q} \rightarrow 0$, the linearized Thomas-Fermi dielectric function. It only assumes linear response, that is, the induced density of charge is proportional to the total potential $\phi(\mathbf{r})$.
Note also that the Lindhard treatment is far more general than the Thomas-Fermi in the sense that it can describe both dynamic and static screening.
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Ideal gas in a vessel: kinetic energy of particles hitting the vessel's wall Reading Landau's Statistical Physics Part (3rd Edition), I am trying to calculate the answer to Chapter 39, Problem 3.
You are supposed to calculate the total kinetic energy of the particles in an ideal gas hitting the wall of a vessel containing said gas.
The number of collisions per unit area (of the vessel) per unit time is easily calculated from the Maxwellian distribution of the number of particles with a given velocity $\vec{v}$ (we define a coordinate system with the z-axis perpendicular to a surface element of the vessel's wall; more on that in the above mentioned book):
$$
\mathrm{d}\nu_v = \mathrm{d}N_v \cdot v_z = \frac{N}{V}\left(\frac{m}{2\pi T}\right)^{3/2} \exp\left[-m(v_x^2 + v_y^2 + v_z^2)/2T \right] \cdot v_z \mathrm{d}v_x \mathrm{d}v_y \mathrm{d}v_z
$$
Integration of the velocity components in $x$ and $y$ direction from $-\infty$ to $\infty$, and of the $z$ component from $0$ to $\infty$ (because for $v_z<0$ a particle would move away from the vessel wall) gives for the total number of collisions with the wall per unit area per unit time:
$$
\nu = \frac{N}{V} \sqrt{\frac{T}{2\pi m}}
$$
Now it gets interesting:
I want to calculate the total kinetic energy of all particles hitting the wall, per unit area per unit time. I thought, this would just be:
$$
E_{\text{tot}} = \overline{E} \cdot \nu = \frac{1}{2} m \overline{v^2} \cdot \nu
$$
The solution in Landau is given as:
$$
E = \nu \cdot 2T
$$
That would mean that for the mean-square velocity of my particles I would need a result like:
$$
\overline{v^2} = 4\frac{T}{m}
$$
Now, I consider that for the distribution of $v_x$ and $v_z$ nothing has changed and I can still use a Maxwellian distribution. That would just give me a contribution of $\frac{T}{m}$ each. That leaves me with $2\frac{T}{m}$, which I have to obtain for the $v_z$, but this is where my trouble starts:
How do I calculate the correct velocity distribution of $v_z^2$?
| The reason your calculation is not right is because the mean energy of the molecules hitting the wall is not the mean number times the mean energy per molecule, because the fast molecules hit the walls more frequently than the slow ones.
| {
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Possibility for radiation in dark matter that is not interactive with regular matter? Definition: Radiation in this case does not refer to electromagnetic radiation. It refers to any kind of emission of energy, even energy that does not interact with regular matter.
Just like dark matter does not interact with electromagnetic radiation, could regular matter not interact with "dark matter radiation" (I'm not talking about the usual "really high wavelength radiation" kind)?
| Well, I claim the answer is yes! Since the question defines radiation as emission of energy that doesn't interact with normal matter, I say that dark matter emitting dark matter particles qualifies as emission of energy that doesn't interact with normal matter. An clearly it can do that.
| {
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Which is the strongest meteor shower expected in the next years in the Northern hemisphere? Which is the strongest meteor shower expected in the next years in the Northern hemisphere?
Is it possible to give good predictions for this?
| The Perseids in August are always good with 30+ meteors per hour. If you can get to a dark sky, you won't be disappointed. The Leonids hit their thirty-three year peak just a couple years ago so it will be a while before they peak again. The Perseids peaked in the mid-1990s (I saw 200 meteors in 1.5 hours through a hole in the clouds that just showed Perseus, Cassiopeia and Pegasus), but I don't know what the period of that shower is.
You can find lots of useful meteor shower information at the American Meteor Shower website.
| {
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What objects look best in an O-III filter? I've heard that an O-III (Oxygen III) filter is great for planetary nebulae.
Is this true for all planetary nebulae, or just some or most?
What other target types are often improved with an O-III filter?
| Most planetary nebulae are improved with an O-III filter. I find it particularly helpful for the large dim Helix Nebula.
I also find it useful for most diffuse nebulae and supernova remnants, especially the Veil Nebula. For some reason, it doesn't help the Crab Nebula at all.
| {
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What is the current status of Pluto? Pluto has been designated a planet in our solar system for years (ever since it was discovered in the last century), but in 2006 it was demoted.
What caused this decision? And is there a chance that it could be reversed?
Edit: well, http://www.dailygalaxy.com/my_weblog/2017/03/nasas-new-horizon-astronomers-declare-pluto-is-a-planet-so-is-jupiters-ocean-moon-europa.html is interesting; this is science, so anything could (potentially) change.
| As many already said, Pluto is now considered a "dwarf planet"
For your second question, there is no chance that Pluto will be reclassified as a planet again.
| {
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Can any telescope be used for solar observing? Can any telescope, such as a 8" reflector, that is normally used at night to look at planets be used or adapted for solar observing?
What kind of adapters or filters are required or is it better to get a dedicated solar telescope?
I'd like to look at sunspots, flares, prominences, eclipses, etc.
| For viewing the moon I stopped down a 10" Newtonian reflector to 2", using tinfoil, thick paper, duct tape and a mason-jar ring (which is very round). I think this might also work for a projecting an image of the sun but with a much smaller aperture. The tinfoil cover with over the aperture, with its hole hopefully will reflect away some of the heat. (from Central Texas where we will see only about 93%, looking forward to 2024!)
| {
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What open-source n-body codes are available and what are their features? I'm interested in doing simulations with large numbers of particles and need a good n-body code. Are there any out there in the public domain that are open-source and what are their strengths and weaknesses. I'm interested in all types of codes, ones that can be run on a multi-core desktop for basic simulations and also ones that can be run on large parallel clusters (I have access to both).
For each entry please provide a link and a brief summary of the nature of the code and strengths of the software. The goal here is to provide a reference list for those interested in the topic.
| Amara Graps wrote a good overview of n-body simulation methods.
| {
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Video of Earth spinning? If the Earth is spinning or rotating at a really fast speed, why haven't we seen any videos from space of it spinning when we get a lot of photos of it?
| More importantly, in order to see the rotation properly, you'd need to stay at the same relative point in space. But in order to orbit the Earth, you typically need to go around it faster (until you get to geosynchronous orbit). The video's from the ISS make the Earth look like it's spinning because the ISS is orbiting so quickly. (It's mean orbital time is about 90 minutes.) I guess if the Apollo astronauts had taken a video recording of the Earth, from the moon for just over a day, you'd see about one full revolution of the Earth. But as said, that wasn't exactly a high priority: it's not like we're in any doubt that the Earth rotates once per day!
| {
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What's the best way to start out with astrophotography on a tight budget? I'm new to astronomy and I don't have a great deal of cash to spend. I currently have a 3" reflector telescope which I've had great fun with. I also have a Pentax DSLR which I've been using to take long exposure photos of various constellations (+ brighter deep sky objects such as the Orion nebula).
I've managed to take some surprisingly good photos of the moon and jupiter's moons combining my mobile phone with my telescope (which requires a very steady hand and alot of patience) but I'd really like a better way of taking photos. I'm wondering whether I should investigate getting an SLR mount to attach my Pentax to my scope, or whether to go for one of these webcam eyepeiece attachments. Any advice?
| Without investing in a better telescope first I think a DIY webcam type setup or an entry level astrocam targeted at planetary use would probably be the best choice. A 3" reflector is unlikely to have a mount capable of tracking smoothly enough for long exposure; and the larger weight of a DSLR will cause more problems due to the asymmetric load that will result from the placement of the focuser on a Newtonian scope. A wobbly mount doesn't matter much when you're taking large numbers of very short exposures (like video).
Alternately, you might be able to use your phones camera directly for this if you can find a tripod adapter for it and combine it with an adapter using a tripod screw to mount a point and shoot camera behind your eyepiece.
| {
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AGN accretion disk vs. torus The torus is the donut of dust encircling the Active Galactic nucleus. The accretion disk is inside the torus. Is there a boundary between the two? At what point does a torus become an accretion disk? What are their differences?
| If I understand the literature right, the disk "ends" when it reaches the self-gravity radius $R_{sg}$ where local gravity from the disk exceeds the vertical component of the black hole gravity and the disk becomes unstable. $$R_{sg}\approx 1680 \left[\frac{M_{BH}}{10^9 M_\odot}\right]^{-2/9} \alpha^{2/9} \left[\frac{L_{AGN}}{L_{Edd}}\right]^{4/9} \left[\frac{\epsilon}{0.1}\right]^{-4/9} R_S$$ where $\alpha$ is the viscosity parameter $\approx 0.01-0.1$. For a $10^9 M_\odot$ black hole with $L_{AGN}/L_{Edd}\approx 0.1$ $R_{sg}\approx 0.04$ pc. This appears to be in rough agreement with observations of quasars.
| {
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Color Variation in RR Lyrae I've been doing some research on RR Lyrae stars and haven't been really able to find an answer to this question.
RR Lyrae are well known for their periodic magnitude, and also are usually found in certain color ranges ("RR Lyrae Color Box"). My question is: does the color of RR Lyrae Stars (G-R in particular) vary over it's period?
I would also love if anyone has a link to a topical journal article!
| Although I'm a regular variable star observer, I don't observe RR Lyrae variables because they are not suitable targets for visual observers like myself. However, the long period Mira-type variable that I observe the most also change colour with brightness. At their dimmest, they look like glowing red coals, but they fade to white as they get brighter. In fact, I can usually spot the variable star in the field because of its striking red colour. Some good examples are Mu Cephei (Herschel's Garnet Star) and R Leporis (Hind's Crimson Star).
| {
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Why did the WMAP mission last so much longer than Planck? NASA endorsed 9 years of data taken with the Wilkinson Microwave Anisotropy Probe (WMAP). The High Frequency Instrument aboard the Planck satellite ran out of coolant at the start of 2012, after about two and a half years of operation. Even if the Low Frequency instrument runs for another two years, its operating life will be much shorter than WMAP's.
Why is there such a big difference between the operational life-spans of the two missions, given that they have basically the same objective: to measure the cosmic microwave background?
| Dan Neely gave an excellent answer. I wanted to add that WMAP was initially planned as a one-year mission, and soon extended to two years---in which it reached its designed sensitivity goals.
After that point, it continued due to its success and the reasons listed by Dan.
| {
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The Sun as a gravitational lens Since the Sun is a gravitational lens with as focal length of 550 AU for visible light, with an immense amplification factor, shouldn't it light up objects hanging out there?
We should get solar sails up there to finally boost them somewhere interesting, opposite to the star that is emitting the light.
For neutrinos, I've heard that the focal length is 110 AU. Wouldn't a massive neutrino bombardement trigger interesting nuclear reactions (maybe that's not an astronomy question)?
| How does 200 metre resolution at Alpha Centauri grab you?
http://www.cesr.fr/~pvb/gamma_wave_2005/presentations/optics/Koechlin.pdf
*
*Wherever in space there are intelligent creatures like us, they
will be driven to explore and understand our universe, just as we
do. We and they wish to see to the farthest depths of space with
the greatest clarity allowed by the laws of nature. To this end, we
build, at great expense, ever more powerful telescopes of all
kinds on Earth, and now in space.
*As each civilization becomes more knowledgeable, they will
recognize, as we now have recognized, that each civilization has
been given a single great gift: a lens of such power that no
reasonable technology could ever duplicate or surpass its power.
This lens is the civilization's star. In our case, our Sun.
*The gravity of each such star acts to bend space and thus the
paths of any wave or particle, in the end creating an image just as
familiar lenses do.
*This lens can produce images which would take perhaps
thousands of conventional telescopes to produce. It can
produce images of the finest detail of distant stars and
galaxies.
*Every civilization will discover this eventually, and surely
will make the exploitation of such a lens a very high priority
enterprise.
*One wonders how many such lenses are being used at this
moment in time to scan the universe, capturing a flood of
information about both the physical and biological realities
of our time.
6
*Frank Drake, 1999, from his Foreward to Claudio’s book.
| {
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What is behind the cosmological horizon barrier? I'm wondering what is behind the cosmological horizon barrier?
| Well - there are a couple of possibilities:
*
*Nothing: the universe could actually be the size we can see, with the edge about 46 billion light-years away
*Lots more universe, similar to what we can see
*An infinite universe
It doesn't really matter which, though, as nothing beyond that horizon can effect us or be affected by us (the possible exception being objects around the same distance as the horizon, which may pop into view, due to variation in the Hubble parameter)
| {
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Why isn't sunset time in sync with solstice? The winter solstice in the Northern Hemisphere in 2011 is on December 22.
But if I look at the sunset times for a location such as Washington, DC on the USNO site, the sunset time starts reversing much earlier (around December 6th).
Shouldn't the sunset time start reversing (from earlier to later) on the same date as the solstice?
Update:
Omega Centauri's explanation is correct.
*
*The Royal Observatory gives a similar account of this oddity on this page under the section titled "The apparently odd behaviour of sunrise/set times near the winter solstice".
*Cornell University provides more details here.
| This is because the earth doesn't have a circular orbit, so some times the angular velocity of the earth is greater than average (and also the reverse). Perihelion is in early January, so the planets orbital velocity is greater (i.e. the Northern hemisphere winter is shorter than half a year long). So the affects the length of a day, making it differer slightly from 24hours depending upon the season. So solar time versus clock times drifts.
| {
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If neutrinos travel faster than light, how much lead time would we have over detecting supernovas? In light of the recent story that neutrinos travel faster than photons, I realize the news about this is sensationalistic and many tests still remain, but let's ASSUME neutrinos are eventually proven to travel "60 ns faster than light". If so, how much lead time would they have over light from local supernovas (e.g. SN 1987A) and distant (e.g. SN 2011fe)?
What does the math look like to calculate this?
| The calculation is done for 1987A here. Basically, the neutrinos' fractional speed increase from the new paper is $2.48\pm0.28\pm0.30\times10^{-5}$ (statistical / systematic errors, respectively) . SN1987a was $166\,912\pm10.1$ ly away, so multiplying the fraction by the travel time gives $4.14\pm0.97$ years. In reality, we got the neutrinos a few hours beforehand, but mostly because the light had to scatter out.
| {
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What was the apparent magnitude of the June 15th 2011 lunar eclipse? My plan was to observe and estimate the apparent magnitude of the Moon during totality of the June 2011 lunar eclipse, but the clouds rolled in at the exact moment, so I couldn't make any useful estimates. What was the apparent magnitude?
| I was on the wrong side of the planet to see this eclipse, but I would expect it was an extremely dark eclipse, because the Moon passed very close to the centre of the umbral shadow.
| {
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Can the Hubble telescope bring any star into focus? Lets say I am talking about a view like this supernova - 13 billions light year away. In short can Hubble bring any star into focus in the entire universe? And if so, to what definition?
I also wonder, how much time would time would it need to focus on a distant star or planet, in comparison to a closer one?
| Once a telescope is set in proper focus, objects at all distances are rendered into sharp images without any further changes in focus settings. However, the lowest luminosity of stars that can be observed does depend on distance. And the farther away a galaxy is, the less detail can be seen in its image.
Hubble can image a star like the Sun about as far as the nearest spiral galaxy, M31, which is about 2.4 million light years away. If the Universe were unevolving and unexpanding, a supernova could be observed by Hubble up to 30 billion light years away. But the Universe expands, shifting visible and near ultraviolet light of the most distant stars to wavelengths longer than those that can be detected by Hubble instruments. Light emitted in the far ultraviolet would be shifted to visible wavelengths, except that this light was absorbed in the very early universe by intergalactic clouds of neutral hydrogen.
The upshot is that the most distant objects visible to Hubble are about 13.2 billion light years away. And no telescope can ever see more than 13.8 billion light years away because that is the distance travelled by light since the Big Bang which occurred 13.8 billion years ago. As for the detail on these most distant objects, it becomes difficult to impossible on Hubble images to separate the image of a supernova from its host galaxy.
| {
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Where do the bipolar jets of black holes come from? How are they formed? And why are they so bright?
| Black holes are small relative to their mass, matter that falls into a black hole will accelerate to very high speeds and then get crammed into a very tight volume. If even a relatively modest amount of material is falling onto a black hole it will be bottlenecked like sand falling through the middle of an hourglass. This process creates an accretion disc of matter that is queued up trying to fall into the black hole. Because of the extreme forces involved the friction in the accretion disc is so strong that it heats matter to extreme temperatures. The accretion process is one of the most efficient in converting rest-mass to energy, even fusion only converts a fraction of a percent of rest-mass to energy, but accretion can convert up to 40%.
These enormous temperatures not only create lots of high energy EM radiation but also ionize atoms and create charged particle/anti-particle pairs. Electrically charged particles in motion will create magnetic fields, in a black hole's accretion disc there are a lot of electrically charged particles moving very, very fast, creating extremely strong magnetic fields, which in turn accelerate ions in jets perpendicular to the disc. The formation of the jets themselves is an enormously complex process (since the ions both create the magnetic field and are accelerated by it) that is not well understood. Nevertheless, the enormous energies involved are very favorable to the production of charged particles and accelerating them to high energies in the bipolar jets. They are so bright because the particles within them are at very high energy (particle/particle collisions will release a lot of energy, recombination of ions with electrons will release a lot of energy) and also because of the presence of large amounts of anti-matter (resulting in annihilation reactions which release gamma rays).
| {
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Why is a new moon not the same as a solar eclipse? Forgive the elementary nature of this question:
Because a new moon occurs when the moon is positioned between the earth and sun, doesn't this also mean that somewhere on the Earth, a solar eclipse (or partial eclipse) is happening?
What, then, is the difference between a solar eclipse and a new moon?
| The Moon's orbit is inclined with respect to the Earth's orbit. In other words, if you imagine a Sun, Earth, and Moon model sitting on a tabletop, the Sun would sit approximately still and the Earth might slide around the desktop, while the Moon would orbit the Earth, hopping up off the table, and sinking back down into it. (I used to do this demonstration with my astronomy labs.)
A new Moon occurs whenever the Moon is merely on the same side of the Earth as the Sun, but it may or may not be on a level with the Sun. In that case, you would not have an eclipse. If, however, the Moon happened to be in the middle of its up-and-down travel at the same time that it crossed the Sun's side of the Earth, then you could get an eclipse if the alignment were precise enough.
Basically, a New Moon is when the Sun and Moon are vaguely in the same direction, while an eclipse occurs when they are in almost exactly the same direction. For a total eclipse, the alignment has to be nearly perfect.
EDIT: The best picture I could find, showing the Moon out of the plane of the Earth's orbit- http://www.phys.ufl.edu/demo/1_Mechanics/L_Gravity/SunEarthMoon.html
| {
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How could I translate a field of view value into a magnification value? When I zoom in with Stellarium, it indicates a field of view (FOV) value in degrees, but most binoculars and telescopes are advertised with value like "nX magnification power."
How could I translate this value so I get an idea of what I will see with a telescope or binocular?
For example, I if got a 30X telescope, how much should I zoom to get similar view?
| Different telescope and binocular eyepieces have different fields of view, so that there is no direct relationship between magnification and field of view.
Eyepieces range in apparent field of view from 30° to 110°, typically being in the range of 50° to 70°. For any given eyepiece, you can calculate the actual field of view by dividing the apparent field of view by the magnification. Thus a 30x eyepiece with a 60° apparent field of view will show you an actual field of view of 60°÷ 30x = 2°.
| {
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Mass of a galaxy via luminosity Is there a general formula for calculating the mass of a galaxy, or even a nebula from the luminosity? Or, is there a way of calculating the total mass of a galaxy from its energy output?
Is there a Hertzsprung–Russell diagram equivalent for galaxies?
I know about gravitational lensing or velocity dispersion via the virial equation, and the Schechter function, and using doppler spread to calculate a mass.
| It sounds like you already have a pretty comprehensive answer in hand, but I would mention that galactic clusters are often classified by the mass or luminosity of their Nth-biggest or brightest member, where N is a smallish integer like 5. The idea is that the biggest couple might be weird outliers, but by the time you get down to the "rank and file" galaxies in a cluster, you should have a good handle on how big the cluster is. The strength of that scheme is that it can be difficult to determine if every galaxy in your image is actually a member, so that it is difficult to arrive at a good estimate of total mass directly. ...And you only have to take detailed data on a handful of the easiest galaxies anyway.
EDIT: I knew there was another, more direct answer, but I couldn't remember the authors' names. The Tully-Fisher Relation and the Faber-Jackson Relation describe empirically the relation between galactic luminosity and velocity-dispersion for spiral and elliptical galaxies, respectively. It is usually used to infer the former from the latter, but if for some reason you had luminosity in-hand already, velocity dispersion is related to the strength of the gravitational field of the galaxy and thus its mass.
| {
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In astronomy what phenomena have theory predicted before observations? As far as I know, astronomy is generally an observational science. We see something and then try to explain why it is happening. The one exception that I know of is black holes: first it was thought of, then it was found.
Einstein's relativity is middle ground to me, he thought of light beams at the speed of light but obviously could observe gravity's effects.
Anyway, I guess my question is, what are the biggest discoveries that were thought of before they were seen in the sky?
| The deflection of light by the sun was first predicted by Einstein's general relativity, then observed in a solar eclipse.
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How to find the Andromeda galaxy without using a go-to telescope? In other words, what is the proper technique (star-hopping or other?) in order to find and properly point a telescope to this target? Would a star atlas or other tool/reference help? Can I use the R.A. and Dec. coordinates to find such deep-space objects?
I can recognize the constellations around which M31 is located, but I could never figure out a way to pin-point it. Any help or explanation is much appreciated.
Thank you!
| Maybe I'm the only one who starts at Cassiopeia, but I find it easy to start in the appropriate "wedge" of Cassiopeia (with Schedar at it's point) and follow that towards the bright line of stars in Pegasus (Mirach, etc. ).
| {
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Why are stars, planets and larger moons (approximately) spherical in shape (like, the Sun, the Moon, the Earth, and other planets)? Why are stars, planets and larger moons (approximately) spherical in shape (like, the Sun, the Moon, the Earth, and other planets)?
| Adding to the previous answer, consider the surface of the Earth. Any deviation from a spherical shape is either an indentation (canyon, valley) or a bump (hill, mountain). Any such deviation that's too big is smoothed out by gravity; a 100-kilometer mountain would collapse or sink, and a 100-kilometer deep valley would be filled in. The result is that there are some relatively small deviations (tiny bumps like the Himalayas), but the overall shape is very nearly spherical.
Rotation causes some large-scale distortion; Earth's equatorial radius is about 20 kilometers greater than its polar radius.
It's the same for any other body large enough to have significant gravity. Small moons and asteroids can be quite irregular, and rapidly spinning bodies can be significantly flattened, but most coherent bodies are very nearly spherical.
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What is exactly the density of a black hole and how can it be calculated? How do scientists calculate that density? What data do they have to calculate that?
| A black hole is a celestial body of extreme density and high gravitational pull that not reflect or emit radiation.
The process of forming a black hole is related to the evolution of some stars. As you know, a star of similar mass to the Sun ends up becoming a white dwarf, a small star with high density.
The explosion of a nova leaves behind a new star of enormous density and small volume with a diameter not exceeding 10 km., Consisting solely of neutrons.
moreover, the density of a black hole should not be the same for all, because each has a different size depending on the original mass of the collapsed star. but that there should be no doubt that this density is very high.
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What is Hawking radiation and how does it cause a black hole to evaporate? My understanding is that Hawking radiation isn't really radiated from a black hole, but rather occurs when a particle anti-particle pair spontaneously pop into existence, and before they can annihilate each other, the antiparticle gets sucked into the black hole while the particle escapes. In this way it appears that matter has escaped the black hole because it has lost some mass and that amount of mass is now zipping away from it.
Is this correct? If so, wouldn't it be equally likely that the particle be trapped in the black hole and the antiparticle go zipping away, appearing as if the black hole is spontaneously growing and emitting antimatter?
How is it that this process can become unbalanced and cause a black hole to eventually emerge from its event horizon and evaporate into cosmic soup over eons?
| You sort of have the answer in your question - but you are assuming mass is positive, as opposed to viewing it as an amount of energy.
Since the particle that is emitted has positive energy, the particle that gets absorbed by the black hole has a negative energy relative to the outside universe. This results in the black hole losing energy, and thus mass.
Smaller primordial black holes can emit more energy than they absorb, which results in them losing net mass. Larger black holes, such as those that are one solar mass, absorb more cosmic radiation than they emit through Hawking radiation.
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Why Aren't Saturn's Rings Clumping into Moons? While reading with my son about how a Mars-like planet collided with the early Earth that resulted in our current moon, it said the initial debris also formed a ring, but that ring ended up getting absorbed by the Earth and the Moon.
I couldn't answer his question then why Saturn still has rings. Shouldn't Saturn's rings be clumping into Moons or getting absorbed by Saturn's gravity?
| The rings are too close to the planet.
There is a certain distance from planets called the Roche Limit. Moons can form outside of this limit, but inside, tidal forces are so great that they prevent particles from forming into moons under their own gravity. For instance, our moon is outside Earth's Roche Limit. Saturn's rings, on the other hand, are inside.
| {
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How much of the Earth would a spoonful of the Sun scorch? How much of the Earth would a spoonful of the Sun scorch if held at ground level?
I basically would like to conceptualize the heat of the Sun on a smaller scale, please.
| 10 million tons of neutrons burning at 15 million degrees it would be something similar to crater lake, ground zero would probably be glass, or a chunck of iron, at 15 million degrees it would instantly melt a city just with the flash. 1 teaspoon weighs 10 million tonnes(average mass of a star)it would be a textbook example of a nuclear event.
| {
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What is this shadow of the Sun on the Moon? I was reading the article Moon Phases on HowStuffWorks. In the picture, each moon has a dark green area which represents the shadow of the Sun. How is this shadow formed and why is this important?
| The half of the Moon not represented by the dark green semicircle is directly illuminated by the Sun. The dark green half is away from the Sun, in solar "night." [Note that this does not mean that it is actually dark there, because the Earth casts a lot of light onto the surface of the Moon as well.] The light and dark areas of Sun illumination determine the phase of the Moon as seen from Earth.
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Does a reference or classification standard for altitude classifications of geocentric orbits exist? I'm looking for a primary reference of the altitude classifications of geocentric orbits (LEO, MEO, GEO, HEO), but I was not able to find something so far.
I noticed that there is very different information about this classification scheme, for example LEO: The English Wikipedia defines LEO from 160 to 2.000 km above mean sea level; in comparison, the German Wikipedia defines LEO from 200 to 1.200 km.
Does anybody know if there is an established standard for this orbit classification, e.g. at NASA, ESA or any other space agency, which can be referenced (f.i. a paper, guideline or handbook)?
Thank you really much!
| NASA's GCMD (Global Change Master Directory) has a classification of orbits which contains the following defintions (truncated for the altitude portion):
*
*LEO : Platforms that orbit between 80 km and 2000 km
*MEO : Platform orbits lie between 2000 km to 35,786 km, but most commonly at 20,200 km or 20,650, with an orbital period of 12 hours).
*GEO : Platform orbits with a revolution of exactly one day at an altitude of 35,786 km
*HEO (High Earth Orbit): any orbit above geosynchronous (above 35,786 km)
*HEO (Highly Elliptical Orbit) : an orbit of low perigee (about 1000 km) and a high apogee over 35,786 km).
They also define LPO (Lagrangian Point Orbits), but they don't define any heliocentric orbits. NASA's Earth Observatory uses the same LEO/MEO/GEO, but seems to lump LPO and Heliocentric into HEO.
And for referencing, see Citing GCMD Keywords
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Counting complete sets of mutually unbiased bases composed of stabilizer states Consider $N$ qubits. There are many complete sets of $2^N+1$ mutually unbiased bases formed exclusively of stabilizer states. How many?
Each complete set can be constructed as follows: partition the set of $4^N-1$ Pauli operators (excluding the identity) into $(2^N+1)$ sets of $(2^N-1)$ mutually commuting operators. Each set of commuting Paulis forms a group (if you also include the identity and "copies" of the Paulis with added phases $\pm 1$, $\pm i$). The common eigenstates of the operators in each such group form a basis for the Hilbert space, and the bases are mutually unbiased. So the question is how many different such partitions there exist for $N$ qubits. For $N=2$ there are six partitions, for $N=3$ there are 960 (as I found computationally).
The construction above (due to Lawrence et al., see below) may be an example of a structure common in other discrete groups - a partition of the group elements into (almost) disjoint abelian subgroups having only the identity in common. Does anyone know about this?
Reference:
Mutually unbiased binary observable sets on N qubits - Jay Lawrence, Caslav Brukner, Anton Zeilinger, http://arxiv.org/abs/quant-ph/0104012
| Here is an answer that should work. I do not currently have access to matlab to check this for anything other than the smallest cases, so you should do that.
First off, I find it easier personally to work in the reduced set of $3^N-1$ stabilizers for N qubits (generating the other from those). Entirely a personal preference, and doesn't change the result here.
So we want to divide the $3^N-1$ possible stabilizers into sets of $2^N-2$ commuting stabilizers, and find how many such divisions are possible.
Define
$\alpha = 2^N-2$ = size of sets
$\beta = \frac{3^N-1}{2^N-2}$ = number of sets.
We now pick our sets from the available stabilizers. The first time, we can pick anything - $3^N-1$ choices. Then we have to pick a commuting set, which we'll come back to. After picking the set, there are $(3^N-1) - \alpha$ stabilizers remaining. We can pick any for the first of the next set. And so on. However, for the final set there is no choice: there will only be $\alpha$ stabilizers left. So the choices for the first entry of each set are
$F = \prod_{k=0}^{\beta-2}(3^N-1) - \alpha k$
Now to pick each set. On average, half the stabilizers remaining to chose from will commute with any given one. So picking the second one, half the remainder will do. So for the first set, we have $((3^N-1) - 1)/2$ choices. The next choice has to commute with both the previous ones, so we have $((3^N-1) - 2)/2^2$ choices. And so on. For the next set, we start with $(3^N-1) - (\alpha+1)$ remaining stabilizers to pick the second entry. So the choices for picking the sets are
$S = \prod_{m=1}^{\beta} \prod_{i=1}^{\alpha-1}( \frac{3^N - (\alpha m + 1) - i)}{2^{i+1}} ) $
So the number of possible partitions is $F.S$ divided by the number of possible ways to permute within the sets x number of sets (PR) and number of ways of permuting the sets themselves (PC):
$PR = \beta.\alpha !$
$PC = \beta !$
So the number of partitions is
$\frac{F.S}{PR.PC}$
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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.
| I think that the place to start for you is the Gfitter package. As I understand they have the best "global fit" of the SM parameters at $m_Z$.
Here is the recent paper which takes into account recently discovered God pa ..., sorry, the Higgs boson.
Also note that they've partially done your work by considering Type-II 2HDM.
Hope that helps.
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Literature on fractal properties of quasicrystals At the seminar where the talk was about quasicrystals, I mentioned that some results on their properties remind the fractals. The person who gave the talk was not too fluent in a rigor mathematics behind those properties, and I was not able to find any clues to this area myself.
There are some papers where fractals are mentioned in application to quasicrystals, but I did not find any introductory level paper with careful mathematics. Probably there is such an introduction somewhere?
| Finally I asked the person working in this field and he gave me the paper http://prb.aps.org/abstract/PRB/v35/i3/p1020_1 where fractality of the spectrum is shown in rather reasonable model.
| {
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What are some ways to (approximately) symbolically diagonalize Hamiltonian operator? Specifically the Hamiltonian takes the form of
$$\hat H = \frac{\Delta }{2}{\hat \sigma _z} + {\omega _1}\hat a_1^\dagger {\hat a_1} + {\omega _2}\hat a_2^\dagger {\hat a_2} + {g_1}\left( {{{\hat a}_1}{{\hat \sigma }_ + } + \hat a_1^\dagger {{\hat \sigma }_ - }} \right) + {g_2}\left( {{{\hat a}_2}{{\hat \sigma }_ + } + \hat a_2^\dagger {{\hat \sigma }_ - }} \right),$$
a three body version of Jaynes-Cummings model.
I'm currently trying to diagonalize this Hamiltonian, a first step in our application of quantum Zeno effect to a three-body system.
I guess this Hamiltonian simply has no close-form diagonalization, just like in classical physics there is no closed-form general solution for a three-body system. So my question is: what are several symbolic approximation techniques to diagonalize an Hermitian operator? Better if that techniques particularly suits this Hamiltonian. The values of $\Delta, \omega_1, \omega_2, g_1, g_2$ need not be general; they can be set, say, all equal in order to simplify calculation.
| At least for $\omega_1=\omega_2$ it is possible to solve the system exactly. Our Hamiltonian can be written
$$\hat H = \frac{\Delta }{2}{\hat \sigma _z} + {\omega}(\hat a_1^\dagger {\hat a_1} + \hat a_2^\dagger {\hat a_2}) + ({g_1}{\hat a}_1+{g_2}{\hat a}_2){{\hat \sigma }_ + } + ({g_1}{\hat a}_1^\dagger+{g_2}{\hat a}_2^\dagger){{\hat \sigma }_ - }$$
We apply a change of variables
$$a_1'=a_1 cos\theta-a_2sin\theta$$
$$a_2'=a_1 sin\theta+a_2cos\theta$$
This change of variables preserves the $\Delta$ and the $\omega$ terms in the Hamiltonian but rotates the $(g_1, g_2)$ vector. By choosing an appropriate $\theta$ we can achieve $g_2'=0$. This means the $(a_1', \sigma)$ system decouples from $a_2'$. The former is an ordinary Jaynes-Cummings model whereas the later is a Harmonic oscillator
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Dual Pairs in Four Dimensions Following the conversation here, I am wondering if anyone knows of an example of dual pair with 4-dimensional N=1 SUSY which relates a non-Abelian gauge theory on one side to a theory with a Lagrangian description but no non-trivial gauge group. Cannot think of one off the top of my head, which doesn't mean it does not exist or even is well-known...
| This feels a little trivial, but I don't see why it isn't an example of what you want: Seiberg duality typically relates an $SU(N_c$) gauge theory with $N_f$ flavors to an $SU(N_f - N_c$) gauge theory. There are degenerate cases when $N_f - N_c = 1$ or $0$, which don't correspond to any dynamical gauge group in the infrared. These are usually described in terms of quantum moduli spaces (s-confining when $N_f = N_c + 1$ and chiral symmetry breaking when $N_f = N_c$), with the low energy fields given by mesons and baryons, but you can equally well describe these as the "dual" quarks and mesons of the usual Seiberg duality, in a degenerate limit without gauge fields coupling to them.
Of course, in the same sense, nonsupersymmetric QCD is dual to a theory of massless pions and no gauge symmetry.
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Group of symmetries of Lagrange's equations Consider the following statements, for a classical system whose configuration space has dimension $d$:
*
*Lagrange equations admit a smaller group of "symmetries" (coordinate change under which equations are formally unchanged) than Hamilton's;
*The 'symplectic diffeomorfism' (=coordinate changes whose jacobian is a symplectic $d$-parametric matrix) Lie group has dimension greater than $\dim G$, $G$ being the (Lie?) group of symmetries of point one.
The first is well known to be true. What about the second? There exists such a $G$ (at a first sight it seemed to me to be the whole $Diff(M)$; but if it is so, then 2 is false)? If it is true, can point 2 explain point 1?
| 0) Let us for simplicity assume that the Legendre transformation from Lagrangian to Hamiltonian formulation is regular.
1) The Lagrangian action $S_L[q]:=\int dt~L$ is invariant under the infinite-dimensional group of diffeomorphisms of the $n$-dimensional (generalized) position space $M$.
2) The Hamiltonian action $S_H[q,p]:=\int dt(p_i \dot{q}^i -H)$ is invariant (up to boundary terms) under the infinite-dimensional group of symplectomorphisms of the $2n$-dimensional phase space $T^*M$.
3) The group of diffeomorphisms of position space can be prolonged onto a subgroup inside the group of symplectomorphisms. (But the group of symplectomorphism is much bigger.) The above is phrased in the active picture. We can also rephrase it in the passive picture of coordinate transformations. Then we can prolong a coordinate transformation
$$q^i ~\longrightarrow~ q^{\prime j}~=~q^{\prime j}(q)$$
into the cotangent bundle $T^*M$ in the standard fashion
$$ p_i ~=~ p^{\prime}_j \frac{\partial q^{\prime j} }{\partial q^i} ~.$$
It is not hard to check that the symplectic two-form becomes invariant
$$dp^{\prime}_j \wedge dq^{\prime j}~=~ dp_i \wedge dq^i $$
(which corresponds to a symplectomorphism in the active picture).
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Examples of heterotic CFTs I'm trying to get a global idea of the world of conformal field theories.
Many authors restrict attention to CFTs where the algebras of left and right movers agree. I'd like to increase my intuition for the cases where that fails (i.e. heterotic CFTs).
What are the simplest models of heterotic CFTs?
There exist beautiful classification results (due to Fuchs-Runkel-Schweigert) in the non-heterotic case that say that rational CFTs with a prescribed chiral algebras are classified by Morita equivalence classes of Frobenius algebras (a.k.a. Q-systems) in the corresponding modular category.
Is anything similar available in the heterotic case?
| The first example that comes to mind is the heterotic string worldsheet theory, described in the original paper of Gross, Harvey, Martinec, & Rohm.
I don't know if there is a classification result for rational heterotic CFTs which generalizes the FRS result. However, if you want to understand the global space of CFTs, you may not want to emphasize rational CFTs anyways. Most CFTs aren't rational.
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Is this a simple Lie algebra? This question comes from Georgi, Lie Alegbras in Particle Physics. Consider the algebra generated by $\sigma_a\otimes1$ and $\sigma_a\otimes \eta_1$ where $\sigma_a$ and $\eta_1$ are Pauli matrices (so $a=1,2,3$). He claims this is "semisimple, but not simple". To me, that means we should look for an invariant subalgebra (a two-sided ideal). The multiplication table is pretty easy to figure out:
$[\sigma_a,\sigma_b]=i\epsilon_{abc}\sigma_c,$
$[\sigma_a,\sigma_b\otimes\eta_1]=i\epsilon_{abc}\sigma_c\otimes\eta_1$
$[\sigma_a\otimes\eta_1,\sigma_b\otimes\eta_1]=i\epsilon_{abc}\sigma_c\otimes1$
I'm dropping off the identity in all the places where it looks like it should be. So the only subalgebra is the $\mathfrak{su}(2)$ generated by $\sigma_a\otimes 1$, and that is not invariant from the second line above. So this looks like a simple algebra to me. Is there a typo somewhere I do not see?
| In this relatively simple example, one can observe that the subalgebras $\{\sigma_a \otimes \frac{1\mp\eta_1}{2}\}$ are the two commuting copies of $su(2)$.
For more complicated situations, one actually has an algorithm to veify the simplicity of a Lie algebra. This is because (the root systems of) simple Lie algebras are classified by Cartan, thus one just needs to verify if the root system of the given Lie algebra coincides with one of the types. Actually, an algebra which is semisimple but not simple will necessarily have orthogonal simple roots.
Taking the given example for illustration. One can choose $\sigma_3\otimes 1$ and $\sigma_3\otimes \eta_1$ as the generators of the Cartan's subalgebra. The corresponding root generators can be found as: $\sigma_{\pm} \otimes \frac{1\mp\eta_1}{2}$. The positive roots can be chosen as: $\alpha = [1,1 ]$, and $\beta = [1,-1 ]$. Now we notice that on one hand they are linearly independent thus they are simple roots.
On the other hand we have $\alpha.\beta=0$ in contradiction to the property of a simple root system requiring that for any distinct simple roots $\alpha.\beta<0$.
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Quantum mechanics as classical field theory Can we view the normal, non-relativistic quantum mechanics as a classical fields?
I know, that one can derive the Schrödinger equation from the Lagrangian density
$${\cal L} ~=~ \frac{i\hbar}{2} (\psi^* \dot\psi - \dot\psi^* \psi) - \frac{\hbar^2}{2m}\nabla\psi^* \cdot \nabla \psi - V({\rm r},t) \psi^*\psi$$
and the principle of least action. But I also heard, that this is not the true quantum mechanical picture as one has no probabilistic interpretation.
I hope the answer is not to obvious, but the topic is very hard to Google (as I get always results for QFT :)). So any pointers to the literature are welcome.
| You certainly couldn't recover quantum effects with a classical treatment of that Lagrangian. If you wanted to recover quantum mechanics from the field Lagrangian you've written, you could either restrict your focus to the single particle sector of Fock space or consider a worldline treatment. To read more about the latter, look up Siegel's online QFT book "Fields" [hep-th/9912205] or Strassler's "Field Theory without Feynman Diagrams" [hep-ph/9205205] for applications of the technique.
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Applications of the Feynman-Vernon Influence Functional I am looking for a reference where the Feynman-Vernon influence functional was defined and used in the context of relativistic quantum field theory. This functional is one method to describe non-equilibrium dynamics for open systems (e.g. coupled to noise) which seems (naively, as an outsider) to be particularly well-suited for field theories where path integral methods are more intuitive.
As a consolation prize, I'd be also interested in applications to other areas of physics (such as dissipative quantum systems), or for more effective or popular methods to describe non-equilibrium dynamics (of open or closed systems) in the context of relativistic quantum field theory (preferably in the path integral language).
| One of the avenues to search for an answer is the so-called Keldysh formalism which is used extensively in condensed matter, in particular in mescopic physics, to define and study steady-state and time-dependent quantum phenomena in systems with infinitely many degrees of freedom. A recent comprehensive review is given by Kamenev and Levchenko, arXiv:0901.3586.
The general idea is as follows: time evolution is defined an a real-time contour going from $t=-\infty$ to $t=+\infty$ and then back, to avoid reference to an unknown final state. The two-time Green functions $G^{ab}(t',t)$ acquire indices $a,b=\pm$ denoting the forward- ($+$) or backward- ($-$) propagating branches of the contour. This gives extra matrix structure to correlators, but many QFT techniques can be adopted to handle this generalization.
I'm not aware of relativistic applications but almost sure it has been done somewhere.
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What are the justifying foundations of statistical mechanics without appealing to the ergodic hypothesis? This question was listed as one of the questions in the proposal (see here), and I didn't know the answer. I don't know the ethics on blatantly stealing such a question, so if it should be deleted or be changed to CW then I'll let the mods change it.
Most foundations of statistical mechanics appeal to the ergodic hypothesis. However, this is a fairly strong assumption from a mathematical perspective. There are a number of results frequently used in statistical mechanics that are based on Ergodic theory. In every statistical mechanics class I've taken and nearly every book I've read, the assumption was made based solely on the justification that without it calculations become virtually impossible.
Hence, I was surprised to see that it is claimed (in the first link) that the ergodic hypothesis is "absolutely unnecessary". The question is fairly self-explanatory, but for a full answer I'd be looking for a reference containing development of statistical mechanics without appealing to the ergodic hypothesis, and in particular some discussion about what assuming the ergodic hypothesis does give you over other foundational schemes.
| I searched for "mixing" and didn't find it in other answers. But this is the key. Ergodicity is largely irrelevant, but mixing is the property that makes equilibrium statistical physics tick for many-particle systems. See, e.g., Sklar's Physics and Chance or Jaynes' papers on statistical physics.
The chaotic hypothesis of Gallavotti and Cohen basically suggests that the same holds true for NESSs.
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Rigorous proof of Bohr-Sommerfeld quantization Bohr-Sommerfeld quantization provides an approximate recipe for recovering the spectrum of a quantum integrable system. Is there a mathematically rigorous explanation why this recipe works? In particular, I suppose it gives an exact description of the large quantum number asymptotics, which should be a theorem.
Also, is there a way to make the recipe more precise by adding corrections of some sort?
| Contrarily to what is generally believed, a semiclassical approximation is achieved through two different series: One is WKB series and the other is the Wigner-Kirkwood series, the latter being a gradient expansion. In both cases, eigenvalues are obtained by the Bohr-Sommerfeld rule but just at the leading order. I have proved this here (this paper appeared in Proceedings of Royal Society A). This proof is rigorous and quite different from what one finds on standard textbooks. Besides, it produces the full series for the exact eigenvalues with at leading order the ordinary Bohr-Sommerfeld rule.
| {
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Question from Schutz's In q. 22 in page 141, I am asked to show that if
$$U^{\alpha}\nabla_{\alpha} V^{\beta} = W^{\beta},$$
then
$$U^{\alpha}\nabla_{\alpha}V_{\beta}=W_{\beta}.$$
Here's what I have done:
$$V_{\beta}=g_{\beta \gamma} V^{\gamma},$$
so
$$U^{\alpha} \nabla_{\alpha} (g_{\beta \gamma} V^{\gamma})=U^{\alpha}(\nabla_{\alpha} g_{\beta \gamma}) V^{\gamma} + g_{\beta \gamma} (U^{\alpha} \nabla_{\alpha} V^{\gamma}).$$
Now, I understand that the second term is $W_{\beta}$, but how come the first term vanishes?
| The covariant derivative is metric compatible, so $\nabla_{\alpha} g_{\beta \gamma} = 0$. This is the condition that the inner product is preserved under parallel transport.
| {
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How to write a paper in physics? I really like to do research in physics and like to calculate to see what happen. However, I really find it hard to write a paper, to explain the results I obtained and to put them in order. One of the reasons is the lack of my vocabulary.
*
*How do I write physics well? I think that writing physics is more dependent of an author's taste than writing mathematics is.
*Are there any good reference I can consult when writing?
*Or could you give me advice and tips on writing a paper?
*What do you take into account when you start writing a paper?
*What are your strategies on the process such as structuring the paper, writing a draft, polishing it, etc?
*In addition, it is helpful to give me examples of great writing with the reason why you think it is good.
*Do you have specific recommendations?
| I will add, that nowadays the introductory part -and even the overall length of the paper- is more important that in classic times. This is because if you paper is too much specialist (and it will be) you must give the reviewers a hint that you have done your homework, that you known your field of study and that you can even give some pointers to guide the revision just in case that the reviewer is not working in your subfield.
| {
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Hilbert-Schmidt basis for many qubits - reference Every density matrix of $n$ qubits can be written in the following way
$$\hat{\rho}=\frac{1}{2^n}\sum_{i_1,i_2,\ldots,i_n=0}^3 t_{i_1i_2\ldots i_n} \hat{\sigma}_{i_1}\otimes\hat{\sigma}_{i_2}\otimes\ldots\otimes\hat{\sigma}_{i_n},$$
where $-1 \leq t_{i_1i_2\ldots i_n} \leq 1$ are real numbers and $\{\hat{\sigma}_0,\hat{\sigma}_1,\hat{\sigma}_2,\hat{\sigma}_3\}$ are the Pauli matrices. In particular for one particle ($n=1$) it is the Bloch representation.
Such representation is used e.g. in a work by Horodecki arXiv:quant-ph/9607007 (they apply $n=2$ to investigate the entanglement of two qubit systems). It is called decomposition in the Hilbert-Schmidt basis.
The question is if there is any good reference for such representation for qubits - either introducing it for quantum applications or a review paper?
I am especially interested in the constrains on $t_{i_1i_2\ldots i_n}$.
| Claudio Altafini studies precisely this subject, in Tensor of coherences parameterization of multiqubit density operators for entanglement characterization and some follow-ups.
| {
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Electricity & Magnetism - Is an electric field infinite? The inverse square law for an electric field is:
$$
E = \frac{Q}{4\pi\varepsilon_{0}r^2}
$$
Here: $$\frac{Q}{\varepsilon_{0}}$$
is the source strength of the charge. It is the point charge divided by the vacuum permittivity or electric constant, I would like very much to know what is meant by source strength as I can't find it anywhere on the internet. Coming to the point an electric field is also described as:
$$Ed = \frac{Fd}{Q} = \Delta V$$
This would mean that an electric field can act only over a certain distance. But according to the Inverse Square Law, the denominator is the surface area of a sphere and we can extend this radius to infinity and still have a value for the electric field. Does this mean that any electric field extends to infinity but its intensity diminishes with increasing length? If that is so, then an electric field is capable of applying infinite energy on any charged particle since from the above mentioned equation, if the distance over which the electric field acts is infinite, then the work done on any charged particle by the field is infinite, therefore the energy supplied by an electric field is infinite. This clashes directly with energy-mass conservation laws. Maybe I don't understand this concept properly, I was hoping someone would help me understand this better.
| It goes out forever, but the total energy it imparts is finite. The reason is that when things fall off as the square of the distance, the sum is finite. For example:
$$ \sum_n {1\over n^2} = {1\over 1} + {1\over 4} + {1\over 9} + {1\over 16} + {1\over 25} + ... = {\pi^2\over 6} $$
This sum has a finite limit. Likewise the total energy you gain from moving a positive charge away from another positive charge from position R to infinity is the finite quantity
$$\int_r^{\infty} {Qq\over r^2} dr = {Qq\over r}$$
So there is no infinity. In two dimensions (or in one), the electric field falls off only like ${1\over r}$ so the potential energy is infinite, and objects thrown apart get infinite speed in the analogous two-dimensional situation.
| {
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Producing photons with same frequency, different amplitude wave I don't understand how two photons of the same frequency can have different amplitudes, neither how to produce them.
I know that classically the square of the amplitude is proportional to the energy, but photons aren't classical particles.
My understanding is that a photon's energy is $h\nu$ - what does the square of its amplitude represent, then? Are there bounds to the amplitude of an EM wave?
Take two waves of amplitudes $A_1$ and $A_2$ and frequency $f_0$. If $A_2 = 2A_1$, can the wave with amplitude $A_2$ be said to carry/be two $A_1$ photons of frequency $f_0$?
| Here is a simple calculation to think about. Imagine a generic radio transmitter emitting, say, 1 watt. That's 1 joule/second. Suppose the frequency is 100 MHz. Take the Einstein relation E=hf, where h is Planck's constant. Figure out the flux of photons (number emitted per second). To speak of photons in a QM sense the number of photons needs to be relatively small. The sea of photons emitted by our transmitter behave collectively as a wave: in effect a Bose-Einstein condensate.
| {
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How to determine n equidistant vectors from point P in three dimensions As an assignment for uni I need to figure out an algorithm that explodes a particle of mass $m$, velocity $v$, into $n$ pieces.
For the first part of the assignment, the particle has mass $m$, velocity of $0$, the particle explodes into 6 equal pieces, and is not affected by gravity.
The problem I'm having is, how do I determine 6 equidistant unit vectors in three dimensions?
Assuming I need an initial vector, a unit vector in the direction of $v$ (the initial particle velocity) will do. In the initial case, where $v$ is $(+0, +0, +0)$, let the initial unit vector be in an arbitrary direction.
Also, how do I determine what speed each sub-particle will have? I know that the sum of sub-particle momentum will be equal to the momentum of the initial particle, and because each particle has the same mass they will each have the same speed... Do I need an explosion Force amount or something?
| Interesting problem. Does the explosion need to be symmetric? For arbitrary $n$ I'm not sure this can be done. For $n$ = 6 just make the vectors point to the vertices of an octahedron.
If the explosion doesn't need to be symmetric I would generate $n$ - 1 random directions, speeds and masses, then add up the total momentum of the fragments and calculate the $n$th vector to balance out the momentum and make the net momentum zero. Now add up the masses, then rescale them to make the total mass of the fragments equal the mass of the original particle. This guarantees that momentum and mass are conserved and should give you a reasonably realistic looking "explosion".
You don't say if there are any restrictions on the energies of the fragments. Once you've conserved momentum and mass you can rescale the velocities to give the fragments any total energy you want.
| {
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Given entanglement, why is it permissible to consider the quantum state of subsystems? Quantum entanglement is the norm, is it not? All that exists in reality is the wave function of the whole universe, true? So how come we can blithely talk about the quantum state of subsystems if everything is entangled? How is it even possible to consider subsystems in isolation? Anything less than the quantum state of the whole universe at once. Enlighten me.
| It is true everything is entangled. However, in the many worlds interpretation, the wave function of the universe splits into many worlds. Typically, each decohered world taken by itself will be far less entangled than the wave function as a whole. While taking the partial trace to a subsystem can lead to a very mixed state when done over the complete wave function, the partial trace taken only over one of the worlds will most likely lead to a less mixed state.
| {
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What kinds of materials contract the most in cold temperatures? I know that water expands in the freezer, but I'm curious about which materials contract in response to cold temperatures --- and most importantly, which ones undergo the most drastic changes?
| EDIT: I misread the question
I see you asked what kind of material contracts the most when you cool it. In this regard, hardly anything beats the ideal gas, whose contraction is about .1% per degree at room temperature. If you want a material, consider a bunch of balloons mushed together with drops of glue, or something microscopic equivalent.
Materials that shrink when heated
One can make up a material which contracts as much as you like with heat, by making long polymers which slightly prefer (energetically) be straight. All such chains prefer to be tangled randomly by entropy, so at high temperature, where the entropy contribution is more important, the polymers contract into tight balls, shrinking the lattice.
Such a polymer should be a very long chain, like a hydrocarbon (but hydrocarbons are very stiff, you want something which is flexible). Then you attach spheres to each other using these polymers to make a lattice. This is not quite the description of a rubber band. The universal reason materials contract at higher temperature is that they gain entropy from reducing their volume.
For a realizable material of this sort, rubber is pretty good, but you could make longer chains in more regular arrangements, with supporting materials in the interstitials making it straight. There is no limit to the engineering potential.
| {
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Would you die if you put your hands on a powerline? You know how birds perch on powerlines without getting electrocuted? What if by some chance that I find myself falling and I grab on one of them? Let's say both of my hands are on the same line, would i get electrocuted?
I am thinking I won't because the current won't rush through me and I won't be part of the circuit - me - powerline.
How does the ground play a role in this? I've heard people say that the ground creates a potential difference, but how? There is only voltage across the powerlines, the pole connecting to the ground is wood, an insulator?
Thanks
| Have you seen the helicopter crews that work on overhead lines approach a live wire. They extend a conducting pole and equalize their voltage with the line. After than it is "safe" to work on the line. As long as there is a conductor connecting them to the wire it is ok. Of course the initial equalization process would kill you if the arc went through your heart instead of the conducting pole.
YouTube video of what I am talking about is seen here.
So in theory, yes it would be safe, if conditions are ideal, and everything else stays way away from you, so you don't arc to a nearby tree or something.
PS. It is never "safe", just "safer".
| {
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Any noise slowly starting to take effects? I am studying a system subject to random noise, or a system driven by some noise, for example, heat flow or wave propagation perturbed by noise. I would like to know if there is a real system where the noise take effects slowly instead from the very beginning.
| It's not clear what you mean by "noise" but if you accept black-body radiation as electromagnetic noise then the average peak and total power output ramps up slowly for many systems including something as simple as a light bulb.
| {
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Example of diffusion process without a gradient A book I was reading stated that diffusion can exist without a gradient of a physical quantity. Heat is an example of diffusion because of temperature gradient and similar is the case of mass flow in chemistry. Can anyone give me an example for a gradient less diffusion process?
| I would guess you mean self diffusion: see http://en.wikipedia.org/wiki/Self-diffusion for details.
Suppose you take an aqueous solution of (for example) salt that is uniform so there are no concentration gradients. There is no net diffusion, but the sodium and chloride ions wander around due to random thermal motion, so if you watch a particular sodium atom it will "diffuse" around in a random walk motion.
| {
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How does position uncertainty change in time? I have an online homework for my Modern Physics class, that requires me to find the uncertainty in velocity and position of a duck. The question is as below:
Suppose a duck lives in a universe in which h = 2π J · s. The duck has
a mass of 2.55 kg and is initially known to be within a pond 1.70 m
wide. (a) What is the minimum uncertainty in the component of the
duck's velocity parallel to the pond's width? (b) Assuming this
uncertainty in speed prevails for 4.10 s, determine the uncertainty in
the duck's position after this time interval.
Now, I got the solution to the first part using the equation
$$
\Delta(x).\Delta(p) = h / 4\pi
$$
This gives me $$ \Delta(v) > 0.1153$$
I assume, the second part of the question would be calculated this way:
$$\Delta(x) = \Delta(v) . t$$
or, $$ \Delta(x) = 0.1153 * 4.10$$
But this shows up as the wrong answer. What am I doing wrong? Wouldn't the uncertainty in the duck's position after time t equal the product of velocity uncertainty and time?
(Disclaimer: Yes, this is a homework question as I mentioned above, but I have tried to solve it and seem to be hitting a wall. I suppose I have some concept wrong. A nudge in the right direction would be appreciated.)
| Following David Zaslavsky hint (see comments under question), I realized what I was doing wrong.
I was solving using
$$
\Delta(x) = \Delta(v).t
$$
This would give me
$$\Delta(x) = 0.47273$$
However, I wasn't taking into consideration the initial uncertainty in position.
Thus the correct was of doing it was,
$$\Delta(x) = \Delta(v).t + \Delta(x_{initial})$$
$$ \Delta(x) = 0.47273 + 1.70 = 2.17273$$
| {
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Derivation of the supergravity action in 11D The Einstein-Hilbert action of general relativity is uniquely determined by general covariance and the requirement that only second derivatives in the metric appear. Yang-Mills theory can be motivated in a similar way. In the original paper of Scherk, Julia, Cremer there are some arguments given from which they deduced the form of the action. They are only sketched however. Is there a more complete exposition of the derivation in the literature, or possibly even a uniqueness result as in the case of general relativity or Yang-Mills theory?
| A beautifully conceptual derivation of 11d SuGra was given in
*
*Riccardo D'Auria, Pietro Fré, Geometric Supergravity in D=11 and its hidden supergroup, Nuclear Physics B201 (1982) 101-140 (nLab)
using the excellent supergeometric methods later laid out in their textbook
*
*Leonardo Castellani, Riccardo D'Auria, Pietro Fré, Supergravity and Superstrings - A Geometric Perspective World Scientific, 1991 (nLab)
| {
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What is a "Center Of Mass" issue of a Gorillapod? I read somewhere that a Gorillapod may have "Center Of Mass" issues when used with the long lenses.
So, I wish to understand what is a "Center Of Mass" issue?
I have to clarify that I am NOT a physics student nor I ever intend to be. Answers in a layman's language would be appreciated.
| Stand on your tiptoes and hold your hands out. You will discover what center of mass issues means. Basically gravity is pushing down on the camera/lens and it needs to be supported directly underneath where gravity is acting (center of gravity).
| {
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Work done on charged particle by magnetic field in quantum mechanics Classically, we know from $\mathbf{F}=q\mathbf{v}\times \mathbf{B}$ that magnetic field does no work on a charged particle. In quantum mechanics, the Hamiltonian of a charged particle in a magnetic field is given by
$$\hat{H}~=~\frac{1}{2m} \left[\frac{\hbar}{i}\vec\nabla - \frac{q}{c}\vec A\right]^2.$$
How can we deduce from this Hamiltonian whether work done is on the particle?
| First, if the Hamiltonian is time-independent, and your Hamiltonian is assuming that $\vec A$ doesn't depend on time, then energy – the Hamiltonian itself – is conserved.
So what we mean by the magnetic field's doing no work is that the kinetic energy of the particle doesn't change i.e. the speed doesn't change. So one must determine what the kinetic energy or speed is.
By the Heisenberg equations of motion (or, by an equivalent classical procedure of relating $L$ and $H$ and the canonical velocities with canonical momenta), the speed may be determined from the commutator of $H$ with $x$ because
$$ i\hbar \frac{d}{dt} \vec x = [\vec x,H]$$
is the Heisenberg equation of motion for $x$. Hats are everywhere. Taking your Hamiltonian, the only building block that refuses to commute with $\vec x$ is $\vec \nabla$. If you apply the Leibniz rule, you will easily see that
$$ i\hbar \frac{d}{dt} \vec x = [\vec x,H] = \frac{1}{m}\left[\frac{\hbar}i \vec \nabla - \frac{q}{c}\vec A(\vec x)\right]$$
so the speed – the rate of change of the position – is given by the operator that has both the nabla as well as the vector potential, in the very same combination you wrote. It follows that the kinetic energy is
$$ E = \frac{mv^2}{2} = H $$
i.e. it is exactly equal to your Hamiltonian. The whole Hamiltonian you wrote is the operator of kinetic energy. It commutes with itself so the kinetic energy is conserved (hint: write the Heisenberg equation of motion for the kinetic energy itself) which means that the magnetic field does no work on the charged particle.
| {
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How can there be a quantum field theory that predicts all particle masses? Say I have a theory with only one (energy) scale, e.g. one given by the fundamental constants
$$\epsilon=\sqrt{\dfrac{\hbar c^5}{G}}.$$
In this case, where I can't compare to something else, is there a way to argue that
$$\epsilon<\epsilon^2<\epsilon^3<\dots\ ?$$
By that reasoning, can there be a (field?) theory, where values are obtained from some expansion like in a path integral (which needs a hierarchy of that sort)?
If you really only need/have a theory with $\hbar, c, G$, how can energies like particle masses be deduced from the theory (instead of being experimental input)? And then if, at best, the theory predicts some mass of a particle $\phi$ to be $m_\phi=a_\phi\dfrac{\epsilon\ }{\ c^2}$, then the number $a_\phi$ must have some geometrical meaning, right?
| Here is my take on it from "history" point of view:
*
*With $c$ we understood that there is "no difference" between space and time and we started to use same units to measure them. Same thing for energy and momentum, magnetic and electric fields, e.t.c.
*Then $\hbar$ appeared and we realized that we could measure energies, momenta, distances and time intervals with same units (like $GeV$s). That the inly fundamental unit we have, right? All other units are just introduced for convenience and can be reduced to $GeV$s.
*Now we want to include $G$ into the picture. But that would mean that we will leave the last unit we had -- we will measure all the energies in Planck units. Meaning that we will work just with "pure" dimensionless numbers.
So your theory with only $c,\hbar,G$ parameters should be essentially a pure mathematical construction, that gives you dimensionless numbers "in Planck units".
These numbers can:
*
*have some geometric nature (as you said),
*be solutions or eigenvalues of some functions or operators (as David Zaslavsky said),
*be just "accidental" numbers like Earth radius and even drift with time (as Xiao-Gang Wen said)
Or some combination (or none) of the above.
| {
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Why $\frac{d}{dt}r_{a}\nabla_{a}U_{ab}+\frac{d}{dt}r_{b}\nabla_{b}U_{ba}=\frac{d}{dt}U_{ab}?$ In classical mechanics for two mass particles $a$,$b$ we assume the symmetric potential arising from $F_{ab}$ and $F_{ab}$ given by $$U_{ab}(r)=-\int^{r}_{r_{0}}F_{ab}(r')dr'$$ and $$U_{ba}(r)=-\int^{r}_{r_{0}}F_{ba}(r')dr'$$
The book mechanics by Florian Scheck gives $$\frac{d}{dt}r_{a}\nabla_{a}U_{ab}+\frac{d}{dt}r_{b}\nabla_{b}U_{ba}=\frac{d}{dt}U_{ab}$$ because $$[\frac{d}{dt}r_{a}\nabla_{a}+\frac{d}{dt}r_{b}\nabla_{b}]U_{ab}=\frac{d}{dt}U_{ab}$$
I am confused how we get the summation form. My derivation goes as follows: Notice $F_{ab}=-\nabla_{b} U_{ab}$. Thus we should have $$\frac{d}{dt}U_{ab}=\frac{d}{dr}*\frac{dr}{dt}U_{ab}=\frac{dr}{dt}\frac{d}{dr}U_{ab}=\frac{dr}{dt}[-F_{ab}]=\frac{d}{dt}[r_{a}-r_{b}][-F_{ab}]=[\frac{d}{dt}r_{a}\nabla_{a}+\frac{d}{dt}r_{b}\nabla_{b}]U_{ab}$$
My simple question is just whether my derviation is correct, for I assume $r=r_{a}-r_{b}$ at here.
| Yes, the $r$-argument really is $r_{ik}:=|r_i-r_k|$, as he writes two pages earlier at the beginning of "Systeme von endlich vielen Teilchen". But then you don't need the force to show the relation, it's just the chain rule, which makes derivatives of $U$ into a two term expression and notice that $U(r_{ik})=U(|r_i-r_k|)=U(|r_k-r_i|)=U(r_{ki})$.
Also, it's not so good, that you write "$\frac{d}{dt}r_{a}\nabla_{a}U_{ab}$" for "$\frac{dr_{a}}{dt}\nabla_{a}U_{ab}$", because it suggests that you mean "$\frac{d}{dt}(r_{a}\nabla_{a}U_{ab})$". Moreover, the books name is not the autors name. And I would change the title to something readable, and by that I don't mean the problem with the total derivative, but a title which is a sentence, not a formula. E.g. "A problem deriving the energy conservation for radial two particle potentials".
| {
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What is behind recoherence? I am quite familiar with the concept of decoherence, and I heard that a system that has decohered could recohere after that, I was wondering what could cause the the coherences that have leaked into the environment to come back to the system. I heard about thermal fluctuations for example, without being able to understand this argument.
Thanks.
| The bold and brilliant pioneer Susskind came up with the causal patch conjecture. The universe is accelerating and will approach de Sitter space. The universe right up to the cosmological horizon is all that exists. Nothing beyond exists. The Hilbert space of the universe is finite in dimensionality, approximately $e^{10^{123}}$ in size. A classical system this size will undergo Poincare recurrence at a timescale of the order of the number of states, but a quantum system with wave function components which can interfere will undergo a recurrence in about $\exp\{e^{10^{123}}\}$. Recoherence will happen by then. You can read up more about it in the brilliant conversations between Simplicio, Sagredo and Salviati at
The Multiverse Interpretation of Quantum
Mechanics. Check out their discussion on "unhappening".
| {
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Rotational speed of a discus I was wondering whether the rotational speed of a discus has any influence on the flight of the discus. Would slowing the rotation or speeding it up change the trajectory in any way or would the flight simply become unstable when slowing down?
| The faster it spins, the greater the aerodynamic side force on it; see Magnus effect.
Also, higher rotation increases the $\mu$ (ratio of edge speed relative to body to airspeed of the body) of the disc; the higher airspeed of the advancing edge relative to the retreating edge creates asymmetric lift & drag. The former would impart a rolling moment, while the latter would impart a moment opposing the in-plane rotation of the discus.
All that said, I doubt either are particularly significant effects.
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Quasi 1D insulators with strong spin-orbital interaction We know that the spin-1 chain realizes the Haldane phase which is an example of
symmetry protected topological (SPT) phases (ie short-range entangled phases with symmetry). The Haldane phase is protected by the $SO(3)$ spin rotation symmetry.
If we change the symmetry, we may obtain other possible SPT phases.
This motivates us to ask the following question:
What are good material examples of quasi 1D insulators with strong spin-orbital interaction?
There are large $U$ Mott insulators and there are small $U$ band insulators. Here, we are interested in both, and like to see examples for both cases.
| I can provide an example for bosonic models.
\begin{eqnarray}
\mathcal{H} & = & \mathcal{K} + \mathcal{T}_\text{soc}
+\frac{U}{2}\sum_{i\tau} \hat n_{i\tau}( \hat n_{i\tau}-1) \nonumber \\
& & + U^{\prime} \sum_i \hat n_{i\uparrow} \hat n_{i\downarrow} + V\sum_{i\tau} \hat{n}_{i\tau}\hat{n}_{i+1\tau} \nonumber \\
& & +V^\prime\sum_{i\tau} \hat{n}_{i\tau} \hat{n}_{i+1\bar{\tau}} -\mu\sum_{i}\hat n_{i},
\label{HSOC}
\end{eqnarray}
where
\begin{equation}
\mathcal{T}_\text{soc} = -\lambda\sum_i(\hat{c}^{\dagger}_{i\uparrow}\hat{c}_{i+1\downarrow} -\hat{c}^{\dagger}_{i\downarrow}\hat{c}_{i+1\uparrow})+h.c.,
\end{equation}
is the spin-orbit coupling, $\mathcal{K}$
is the hopping term.
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Electrostatic Pressure Concept There was a Question bothering me.
I tried solving it But couldn't So I finally went up to my teacher asked him for help . He told me that there was a formula for Electrostatic pressure $\rightarrow$
$$\mbox{Pressure}= \frac{\sigma^2}{2\epsilon_0}$$
And we had just to multiply it to the projected area = $\pi r^2$
When i asked him about the pressure thing he never replied.
So what is it actually.Can someone Derive it/Explain it please.
| I think you can do this by dimensional analysis. I'll do the calculation because your professor almost certainly won't accept it, so it's not cheating :-)
We have the three quantities 1/$\epsilon_0$, $\sigma^2$ and $R^n$, where we don't know $n$, and the product has to have the dimensions of force. The dimensions are:
$1/\epsilon_0 = L^3MT^{-4}A^{-2} = L^3MT^{-4}Q^{-2}T^2 = L^3MT^{-2}Q^{-2}$
$\sigma^2 = Q^2L^{-4}$
$R^n = L^n$
and of course force has dimensions $MLT^{-2}$. Multiply together $1/\epsilon_0$, $\sigma^2$ and $R^n$ and set the dimensions equal to $MLT^{-2}$ and you get:
$L^3MT^{-2}Q^{-2}Q^2L^{-4}L^n = MLT^{-2}$
which simplifies to:
$ML^{-1}L^nT^{-2} = MLT^{-2}$
and therefore $n = 2$ and the answer is A.
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Rotationally invariant body and principal axis Suppose a rigid body is invariant under a rotation around an axis $\mathsf{A}$ by a given angle $0 \leq \alpha_0 < 2\pi$ (and also every multiple of $\alpha_0$).
Is it true that in this case the axis $\mathsf{A}$ is a principal axis of the rigid body?
If so, how to prove it? Do you have any references for a proof?
| As a starting point, Wikipedia says on this issue:
The principal axes are often aligned with the object's symmetry axes. If a rigid body has an axis of symmetry of order $m$, meaning it is symmetrical under rotations of 360°/$m$ about the given axis, that axis is a principal axis. When , the rigid body is a symmetrical top. If a rigid body has at least two symmetry axes that are not parallel or perpendicular to each other, it is a spherical top, for example, a cube or any other Platonic solid.
But it gives no canonical reference.
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Gauge invariant but not gauge covariant regularization I'm not sure if someone's already asked this before, but I was wondering, in field theory,
*
*when we say that a certain field is gauge invariant but not gauge covariant, what does this mean? In particular, in Wikipedia, the regulator of Pauli-Villars is said to be as such.
*Moreover, as a consequence of not being gauge covariant, the Wikipedia article says that this regulator can't be used in QCD. How to see the link between not being gauge covariant and QCD here? And, why can one use it in QED then?
| There are gauge conditions that are very useful but are not covariant. Working for example in Coulomb/Transverse gauge you can obtain gauge invariant results that are not manifestly covariant - this is called Gupta-Bleuler approach in QED. The BRS scheme is another example in non-Abelian gauge theories. In such situations Poincare symmetry is restored by restricting the space of available states appropriately.
About Wiki (often wrong) on Pauli Villars: there is no problem creating a gauge invariant and Lorentz covariant scheme, just lots of computations and theoretical complications as gauge invariance arises in combinations of several terms from different order of perturbation theory, a normal feature in any higher order theory including "normal" scalar particle.
I cannot imagine anyone doing PV in QCD.... that is like converting a VW into race car, will work if you must win a bet.....
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Cosmic Background Radiation: How did planets form before the CBR could reach us? I've understood that the Cosmic Background Radiation(CBR) is an electromagnetic wave that originated from the big bang. However, we now live on a planet which that is also originating from the big bang.
Why does that Cosmic Background Radiation reaches us now? Why does CBR reaches us now, and not a couple of billion years earlier?
| The cosmic background radiation was always with us, it is not reaching us now. It just became cooler and cooler as time went on.
One has to understand that when we are talking big bang and general relativity we are talking of the universe starting from one (x,y,z,t) point and as time goes on, expanding. This means that all (x,y,z) points of our universe trace back to that one point singularity that went "bang".
Envision the two dimensional surface of a balloon, as shown in the wiki link. At time=0 the balloon is one point, call it r=0. As it expands the points on its surface start receding from each other, and all points on that surface were at r=0 at t=0. Their neighborhood expands as the balloon blows up, and this means the electromagnetic radiation that started in the earth's neighborhood hot, cools due to the expansion and becomes the Cosmic Microwave Background. That is what I mean it is not coming from anywhere, it just is.
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Determine the acceleration and angular acceleration of a disc The question is:
A 90kg disc is floating in a frictionless vacuum. A 150N force is
applied to the outer rim of the disc. The disc has a radius of 0.25m
and a radius of gyration of 0.16m. What is the acceleration and
angular acceleration of this disc?
To solve it I set up these equations:
\begin{equation}
\mathbf{F}_{a} = \mathbf{m}\times\mathbf{a}\tag{1}
\end{equation}
\begin{equation}
\mathbf{F}_{\alpha} = \frac{\mathbf{I}\times\mathbf{\alpha}}{0.25}\tag{2}
\end{equation}
\begin{equation}
\mathbf{F}_{a}+\mathbf{F}_{\alpha} = 150 N\tag{3}
\end{equation}
You can find I and plug it in along with m, but that still leaves 4 unknowns and only 3 equations. I need a 4th equation but I'm not sure what else is known about the problem.
| There is only one force applied to the system $F = 150\,{\rm N}$ not two, $F_a$ and $F_\alpha$.
Then you have
*
*The total force applied equals the mass times the acceleration of the center of gravity.
*The total torque applied equals the mass moment of inertia (at the center of gravity) times the angular acceleration plus the gyroscopic forces (which are zero in your case).
So what is the torque applied when $F$ is located at the edge of the disk?
What is the mass moment of inertia of disk of mass $m$ with radius of gyration $\rho$?
Once you answer the above questions you can proceed to solve your problem.
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Would the rate of ascent of an indestructible balloon increase as function of it's altitude? Assume a balloon filled with Hydrogen, fitted with a perfect valve, and capable of enduring vacuum (that is to say, it would retain it's shape and so well insulated that the extremes of temperature at high altitudes and in space would have little effect) were to be launched.
As long as the balloon were in atmosphere it would ascend upwards (and also affected by various winds/currents, and gravity). As the balloon passed through increasingly rare atmosphere, would it rise faster?
| Short Answer: NO
Long Answer: the rate of ascent of a balloon is based on his bouyancy (the link mention a fluid, in this case the air), the bouyancy is independent is the gravity field, with density being the only variable.
$$m=\rho_fV_{\mathrm{disp}}$$
reference: wikipedia
with the law of ideal gas the can obtanin the relationship with the presure, and the altitude
$$\rho=\frac{p}{R_{\mathrm{specific}}T}$$
reference: other wikipedia
$$p=p_0\cdot\left(1-\frac{Lh}{T_0}\right)^{\frac{gM}{RL}}\approx p_0\exp\left(-\frac{gMh}{RT_0}\right)$$reference: another wikipedia
this result in the following exponential graph:
Referenceengineering Toolbox
*
*Conclusion:
As Altitude rise, the pressure lowers and proportionality the density, witch lowers the bouyancy net force.
Lastly (cite wikipedia)
"As a balloon rises it tends to increase in volume with reducing atmospheric pressure, but the balloon's cargo does not expand. The average density of the balloon decreases less , therefore, than that of the surrounding air. The balloon's buoyancy decreases because the weight of the displaced air is reduced. A rising balloon tends to stop rising. Similarly, a sinking balloon tends to stop sinking."
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How is external pressure transmitted when a fluid is not enclosed? Pascal's principle states that pressure applied to an enclosed fluid is transmitted undiminished to every part of the fluid.
How is the pressure transmitted in the case where the fluid is not enclosed?
For instance, pushing a block of wood to a beaker with water such that the area of the bottom of the block of wood is does not enclose the mouth of the beaker.
How do we solve for the force on the walls of the beaker?
| I'm not an expert in fluid physics, but after a bit of thought, I believe the following can give clues to your answer.
If the volume of fluid is not enclosed, it means any pressure applied to it that exceeds atmospheric pressure (as in, any amount at all by the block) will just make it flow to the parts of the beaker where the fluid is free to move.
The force applied to walls of a beaker will be a function of the amount of fluid that is over that particular point on the wall - hence why dams are constructed thicker at their bottoms than at their tops. The water at the top of the volume will be at zero pressure relative to the atmosphere (at depth almost 0 the wall can have thickness almost zero and still hold the water). The water at the bottom of the beaker will have a column of water over it - the size of which will be affected by how much volume the block is currently occupying. I believe every 10 or so meters of water equals 1 atmosphere or about 100Kpa.
I'd need to review my textbooks to be able to do real calculations - and currently don't have time for it, but hopefully these are some good starting points for your investigation.
Good luck!
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Show that Bell states cannot be decomposed as tensor products of single qubits' states I'm trying to learn about the Bell state $\frac{1}{\sqrt{2}}|00\rangle+\frac{1}{\sqrt{2}}|11\rangle$. Question 10.1 in Algorithms asks us to show that this cannot be decomposed into the tensor product of two single qubits' states.
It seems to me however that this can be decomposed while still obeying the basic rules. Wolfram Alpha lists some solutions. What am I doing wrong?
| All you need to show is that there do not exist any $a,b,c$ or $d$ satisfying the conditions
$\frac{1}{\sqrt{2}}(|00\rangle)+|11\rangle)=(a|0\rangle+b|1\rangle)\otimes(c|0\rangle+d|1\rangle)$
and $|a|^2+|b|^2=|c|^2+|d|^2=1$.
You should be able to do this quite easily!
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Topology needed for Differential Geometry I am a physics undergrad, and need to study differential geometry ASAP to supplement my studies on solitons and instantons. How much topology do I need to know. I know some basic concepts reading from the Internet on topological spaces, connectedness, compactness, metric, quotient Hausdorff spaces. Do I need to go deeper? Also, could you suggest me some chapters from topology textbooks to brush up this knowledge. Could you please also suggest a good differential geometry books that covers diff. geom. needed in physics in sufficient detail, but not too mathematical? I heard some names such as Nakahara, Fecko, Spivak. How are these?
| My favourite book is Charles Nash and Siddhartha Sen Topology and geometry for Physicists. It has been clearly, concisely written and gives an Intuitive picture over a more axiomatic and rigorous one.
For differential geometry take a look at Gauge field, Knots and Gravity by John Baez. Its very well written starts from the very basics and works its way upwards.
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Do the rings in Mass Effect's mass relays (2-axis gimbal) describe a stable rotation? Just out of curiosity. In the game Mass Effect, devices called mass relays contain two rotating rings, one inside of the other. See http://www.youtube.com/watch?v=qPxw5QjxhIs for an example, best seen around 00:10.
I was wondering: is this a stable motion? Intuitively, I'd say it isn't. Obviously, the outer ring describes normal rotational motion, but when the inner ring is taken into consideration, it seems to me that an additional driving force is required to maintain the entire situation. Am I right? I've been trying to apply some mechanical principles to it, but had no luck so far... Could anyone give a decent mathematical description of this?
Richard Terrett pointed out that this is in fact called a 2-axis gimbal. I wasn't aware of this, thanks!
| According to Euler's rotation theorem, simultaneous rotation around more than one axis at the same time is impossible. If two rotations are forced at the same time, a new axis of rotation will appear. You can read more about this theorem over here: Wikipedia Link
Hope it helps.
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Collision of a black hole & a white hole A black hole and white hole experience a direct collision.
What happens? What shall be the result of such a collision?
| A black hole pretty much is the same as a white hole.
Hawking's result proves they're essentially the same object, so the result will be a black hole with a radius larger than the sum of the radius of the black hole and the "white hole".
I'm just an undergraduate so possibly one of the other members can give a more detailed answer.
edit: I implied but did not directly say that due to a white hole being the same as a black hole your question becomes "what is the result of the collision between two black holes" so the answer is what I said above. I put white hole in quotations because it's really just another (possibly smaller or larger) black hole.
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Number of bits needed to express physical laws? What is the minimum number of bits that would be needed to express a given physical law, like the law of universal gravitation? How many bits are needed to express each of the four fundamental forces? Is there a pattern here?
| One productive way of thinking of the complexity of physical laws is in terms of the Kolmogorov complexity of the algorithm that simulates a given physical situation. This is defined as the length of the shortest code which does the simulation.
If you are given a law of nature, like Newton's law of universal gravitation, you can write a simulation of N-interacting particles. If you are only interested in an in-principle answer, you are looking for the best algorithm to simulate Newton's laws on a computer.
The problem of computing the Kolmogorov complexity precisely is in general the worst of all uncomputable problems. You can usually shrink a code written from scratch by a lot by cleverly rewriting the subroutines to make a specialized language for the description. You would never know if you have the optimal coding, since maybe you could compress things more by adding a special interpretation layer.
The rigorous version of this annoyance is the theorem that no axiomatic system can prove an algorithm is optimal (meaning of minimal length) if the length is significantly greater than the length of the program that does the deduction in this axiomatic system. The proof is a simple contradiction: suppose the axiomatic system proves program P is optimal. Write a program CONTRADICTION which goes through all the theorems of the axiomatic system until it finds a program which is proved optimal and whose length is greater than the length of CONTRADICTION. Then, run this program. By construction, CONTRADICTION is shorter than this program and yet has the same output.
If you think about what CONTRADICTION is doing, it is generating the code for a completely different program, and then running it. This gives a hint of the nature of the difficulty in finding a minimal description.
But if you are happy with a crude estimate of the complexity, you just write any old code to simulate the physics, and the length of this code is an estimate of the complexity. This is a useful heuristic that makes precise the desire for simple elegant theories.
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What are electromagnetic fields made of? I am trying to understand electromagnetic fields so I have two question related to them.
*
*What is a electromagnetic field made of? Is it made of photons / virtual photons?
*How about a static electric or magnetic field?
| Electromagnetic fields, which include static electric and magnetic fields, are indeed made of photons. From a particle physics perspective the Quantum Electrodynamics as a model of particles carrying electric charge interacting via photons has a spectacular agreement with experiment. The thing is, those experiments are very special in that we are sending in 'free' particles with a ton of energy and treating the interactions with the electromagnetic field as a very small perturbation on the free particles. So the picture we draw in our heads of particles interacting via exchange of a single photon is a simplified case that works very well in this situation:
Now, to make the answer more precise for something like a static electric field, to my knowledge is pretty much impossible. To see this we can look at something much simpler, coherent states (see http://en.wikipedia.org/wiki/Coherent_states) . These states don't even have a well-defined photon number, so while they are clearly ' made' of photons as the state is a linear combination of states of well-defined photon number:
$ |\alpha \rangle = e^{ \frac{- |\alpha|^2}{2}} \displaystyle\sum\limits_{n=0}^\infty \frac{\alpha^n}{\sqrt{n!}}|n \rangle $
the the probability of detecting n photons is:
$ P(n) = e^{-|\alpha|^2} \frac{|\alpha|^{2n}}{n!}$
which clearly isn't a delta function for n, which it would be if n was always the same number.
And as far as I can tell, a state which produces a Coulomb-type field ($\frac{k q}{r^2}$) is going to be even more complicated than the coherent states, so it seems hopeless to try and phrase it in these terms. Note that this is in stark contrast to say, the electron number, which is always well-defined. Thus thinking about an electromagnetic field as made up of photons as the same way a block of metal is made up of electrons and other particles is probably a bad analogy to stretch very far.
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Path traced out by a point While studying uniform circular motion at school, one of my friends asked a question:
"How do I prove that the path traced out by a particle such that an applied force of constant magnitude acts on it perpendicular to its velocity is a circle?" Our physics teacher said it was not exactly a very simple thing to prove.
I really wish to know how one can prove it.Thank you!
| One can prove it in a more-or-less elementary way by solving a pair of simultaneous differential equations. In two dimensions, a vector that is perpendicular to a velocity $$\left(\begin{matrix}u(t)\cr v(t)\end{matrix}\right)\quad\mathrm{is}\quad\left(\begin{matrix}-v(t)\cr u(t)\end{matrix}\right).$$ The acceleration, the time derivative of the velocity, is proportional to this vector, so we have the two differential equations
$$\left(\begin{matrix}\dot u(t)\cr \dot v(t)\end{matrix}\right)=\lambda\left(\begin{matrix}-v(t)\cr u(t)\end{matrix}\right).$$
If $\lambda$ is negative, the circle goes "the opposite way". If $\lambda$ is zero, the circle is a straight line.
If you know differentiation and how to solve differential equations, you should be able to solve this pair of equations, and then integrate it to obtain the way in which the position changes over time. If you don't, then it may be better to be patient and wait until you come across it in the course of your studies. Learning calculus on your own to the level needed to solve this differential equation is possible, however.
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Pressures Necessary for Carbon Detonation Carbon detonation is a characteristic event of Type 1a Supernova, where an accreting white dwarf near the Chandrashankar limit of 1.4 solar masses explodes, which is an extremely important standard candle for cosmology. An area of active research is designing computer simulations to model supernova spectra and light curves and fit these to ones obtained observationally to better understand the effect of trace elements and characteristics of the explosion (asymmetry, companion star properties, etc) in order to provide better distance estimates to get more accurate constraints on cosmological parameters (Hubble constant, Dark Energy equation of state, etc).
But it would be very interesting (and extremely cool) if there was a way of generating carbon detonations in a laboratory situation in order to study these effects. What sort of temperature/pressure range is necessary to generate a carbon detonation? Would it be in the range of experimental apparati? Or, on the extreme sides of things, a large thermonuclear device?
| I'm finding two differing definitions online.
In either case, I think that we can say that the full process is a thermonuclear explosion (and hotter, more energetic one than hydrogen/helium ones we use for bombs at that). That is scary bad.
The individual carbon-carbon fusion events can be simulated in a particle physics context with a medium energy ion accelerator, but the resulting data is a long, long way from giving you the full picture of the process you wish to investigate.
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Why isn't it allowed to use a flash when taking pictures in a certain place? When I go to, for example, a museum I try to take some pictures.
Sometimes the museum staffs forbid me to use a flash. Do you know the reason? I don't think it is related to photo-electric effect, right?
| Yes, as you guessed it is related to the photoelectric effect. The photons from the flash, certain frequencies there, can change the molecular composition of the surface paints and pigments.
The precaution is the same as in avoiding a valuable painting or rug to be illuminated by the sun. It is the photoelectric effect.
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What's the difference between Fermi Energy and Fermi Level? I'm a bit confused about the difference between these two concepts. According to Wikipedia the Fermi energy and Fermi level are closely related concepts. From my understanding, the Fermi energy is the highest occupied energy level of a system in absolute zero? Is that correct? Then what's the difference between Fermi energy and Fermi level?
| If you consider a typical metal the highest energy band (i.e. the conduction band) is partially filled. The conduction band is effectively continuous, so thermal energy can excite electrons within this band leaving holes lower in the band.
At absolute zero there is no thermal energy, so electrons fill the band starting from the bottom and there is a sharp cutoff at the highest occupied energy level. This energy defines the Fermi energy.
At finite temperatures there is no sharply defined most energetic electron because thermal energy is continuously exciting electrons within the band. The best you can do is define the energy level with a 50% probability of occupation, and this is the Fermi level.
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Capillaries in series The velocity of fluid of viscosity $\eta$ through a capillary of radius $r$ and length $l$ at a distance $x$ from the center of the capillary is given by; $v=\frac{P}{4l \eta }(r^2-x^2)$ (where $P$ is the pressure difference at the two ends of capillary). With the help of this I can find the rate of flow of fluid out of the capillary equal to $\frac {dV_{out}}{dt} = \frac{\pi Pr^4}{8l \eta }$.
But what happens when the capillaries are in series with different radius and length?
| Assuming the fluids are incompressible, the flow through each capillary must be the same. Also the sum of the pressures across each capillary must equal the total pressure. Therefore, you have the equations:
$P_1+P_2 = P$
$V_1 = \frac{\pi P_1 r_1^4}{8 l_1\eta} = V_2 = \frac{\pi P_2 r_2^4}{8 l_2\eta}$
Solve this system for $P_1$ and $P_2$ then plug back in to find the flow rate in terms of $P, r_1, l_1, r_2, l_2$.
| {
"language": "en",
"url": "https://physics.stackexchange.com/questions/30980",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
What does spin 0 mean exactly? I heard two definitions:
*
*Spin 0 means that the particle has spherical symmetry, without any preferred axis.
*The spin value tells after which angle of rotation the wave function returns to itself: $2\pi$ / spin = angle. Therefore, spin 1/2 returns to itself after $4\pi$, spin 1 after $2\pi$, and spin 0 after an infinite rotation angle.
I also understand that the formula for the return angle is valid for spin 2: such systems return to their original state after a rotation by $\pi$.
These seem to contradict each other: a sphere returns to itself even after infinitesimal rotations. Can somebody clarify?
| Spin zero just means scalar field quanta. No intrinsic angular momentum.
| {
"language": "en",
"url": "https://physics.stackexchange.com/questions/31119",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "22",
"answer_count": 4,
"answer_id": 3
} |
Coriolis Effect and the Space Shuttle The Coriolis effect is a well-known phenomenum, important in meteorology and ocean current forecasting. In addition to location (latitude), it depends on velocity and duration. I assume that commercial aircraft autopilot inertial guidence systems have the ability to compensate for Coriolis, and that even intercontinental missiles are designed with guidance systems that provide Coriolis capability for target accuracy. Was it necessary to provide space shuttles a means to deal with the Coriolis effect during re-entry?
| I am not an expert either, but I think the answer is still simpler than the one that James provided. First of all, the Coriolis effect is only mysterious if you think that your spot on the earth is stationary with respect to say, the sun. Once you acknowledge that a spot on the earth is moving you can take this into account with cannonballs and airplanes alike (they are both trying to hit moving targets). That is, we know that an airplane shouldn't fly from Miami to NY city and shoot for where NY currently is when it leaves Miami, but where it will be in the approximate time it will take to get to the latitude of NY city. For cannonballs we only get to make the choice once, when we shoot it, but for airplanes we can constantly adjust. HOWEVER - if we shoot for where NY city will be in a given time instead of constantly adjusting, we will have a much more efficient flight path that if we constantly adjust to where NY is instantaneously. Thats not to say airplanes don't adjust their flight paths, they can and do, but I would say they use this for instantaneous and evolving flight conditions, rather than something they can easily anticipate - NY city will have 'moved' by the time the flight gets there.
| {
"language": "en",
"url": "https://physics.stackexchange.com/questions/31161",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
Dimension of vector resulting from tensorial product I'm quoting what I found in a book about quantum computation:
Individual state spaces of $n$ particles combine quantum mechanically through the tensor product. If $X$ and $Y$ are vectors, then their tensor product $X\otimes Y$ is also a vector, but its dimension is $\dim(X) \times \dim(Y)$, while the vector product $X\times Y$ has dimension $\dim(X)+\dim(Y)$. For example, if $\dim(X)= \dim(Y)=10$, then the tensor product of the two vectors has dimension $100$, while the vector product has dimension $20$.
I don't understand: how can he state that the result of a vector product has dimension $\dim(X) + \dim(Y)$? What does he intend for dim?
| OP's quote seems to originate from slide p. 45 in Dan Cristian Marinescu's keynote talk from the Computing Frontiers 2004 conference.
Individual state spaces of $n$ particles combine quantum mechanically through the tensor product. If $X$ and $Y$ are vectors, then their tensor product $X\otimes Y$ is also a vector, but its dimension is $\dim(X) \times \dim(Y)$, while the vector product $X\times Y$ has dimension $\dim(X)+\dim(Y)$. For example, if $\dim(X)= \dim(Y)=10$, then the tensor product of the two vectors has dimension $100$, while the vector product has dimension $20$.
The quote mixes the notion of a vector space and the notion of a vector living in that vector space. In particular, it confusingly speaks of a vector product, where it should have referred to a Cartesian product.
Clearly, slides are no substitute for a good textbook. Keep in mind that the speaker may have oversimplified certain points because he didn't need it later in the talk, and that he might have left out less important words on the slides, so that he could use bigger fonts.
Below we suggest a remedy marked in red.
Individual state spaces of $n$ particles combine quantum mechanically through the tensor product. If $X$ and $Y$ are vector $\color{red}{\it spaces}$, then their tensor product $X\otimes Y$ is also a vector $\color{red}{\it space}$, but its dimension is $\dim(X) \times \dim(Y)$, while the $\color{red}{\it Cartesian}$ product $X\times Y$ has dimension $\dim(X)+\dim(Y)$. For example, if $\dim(X)= \dim(Y)=10$, then the tensor product of the two vector $\color{red}{\it spaces}$ has dimension $100$, while the $\color{red}{\it Cartesian}$ product has dimension $20$.
| {
"language": "en",
"url": "https://physics.stackexchange.com/questions/31354",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
} |
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