Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
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Can facial surgical mask effectively filter particles of smoke pollution? Singapore is having an episode of its regular hazy season because of smoke from forest fires. Some people have been using surgical masks like the the one shown at the bottom of this post, presumably for health reasons.
I know the mask has to stop... | Many air inhaled are leaked in, skipping the filter.
The boundaries of the mask do not fit faces, there are crevices. The air leak is huge.
Let's say there are 20% air leak, then 20% virus remained in the air inhaled.
You can test the air leak by pressing the crevices of the mask and inhale, a difference will be felt.
... | {
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How to promote algebraic expressions to operators in quantum mechanics? Okay, I know that in quantum mechanics the quantum observable is obtained from the classical observable by the prescription
$$ X \rightarrow x,\quad P \rightarrow -i\hbar\frac{\partial}{\partial x} $$
in the position basis. Now my questions are: ... | The general problem of converting classical expressions to quantum operator ones is in general unsolvable because classical mechanics is an approximation to quantum mechanics and not the other way around. There is always an ambiguity in how to order noncommuting operators. You have to handle it on a case by case basis,... | {
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Total angular momentum of deuteron I'm studying for my nuclear physics exam and the book we use is Introductory Nuclear Physics by K.S. Krane. In the chapter on Basic Nuclear Structure, we research the deuteron. However, when discussing the total angular momentum of the deuteron, something confuses me (p84 if someone h... | $ 3 \otimes 3 = 1 \oplus 3 \oplus 5$
Where $3$ means "triplet", or a $l=1$ state ($m\in [-1, 0, 1]$). When you combine 2 triplet states, the eigenstate of total angular momentum can be singlet ($l=0$), triplet ($l=1$)--this is the one we're interest in, and "5"-which is $l=2$.
| {
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Simulating an Optical Lens An optical lens focuses the light according to a rule (due to its special shape) and we know that rule. Now, instead of making bigger and bigger lenses, can we use an array of optical sensors and the above mentioned rule to focus the light without actually using a lens?
| What kind of sensors are you talking about? CCDs? You could use a number of CCDs to capture the image and then add the images together in software and map the points to a magnified image which you then project.
Of course, depending on your application, a lens would be cheaper!
| {
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If the multiverse theory is true, can there be a Universe where there are different laws of physics? This is probably a very difficult question. But my question is essentially this, if there are other Universes can different laws of physics exist in those Universes and if so, can't there be a Universe where the laws of... | If there is a "metaverse" in which there are possibly an infinity of independent "Universes", then logically, the existence of any individual "Universe" is contingent on the existence of the "metaverse" rather than the other way around.
That is to say, the laws of any "Universe" must logically be compatible with the ex... | {
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About the microscopic form of magnetocrystalline anisotropy Currently people write magnetocrystalline anisotropy as
$H_{an}=-K s_x^2$
from its classical counterpart:
$H_{an}=-K ( \sin \theta)^2$
where $K$ is the anisotropy constant, but for spin 1/2, $s_x^2$ is just the identity matrix, which shows no anisotropy, so ho... |
I also heard that magnetocrystalline anisotropy comes from spin-orbit interaction, is there some paper depict there relations?
That is correct, and yes. The paper you seek is Tôru Moriya's classic "Anisotropic Superexchange Interaction and Weak Ferromagnetism" [Phys. Rev. 120, 91–98 (1960)]
Your observation is correc... | {
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What is the physics behind a soap bubble? A soap bubble is an extremely thin film of soapy water enclosing air that forms a hollow sphere with an iridescent surface.
What fluid dynamical process occurs during the popping of a soap bubble?
| If you look to Slow-Motion you will find the popping of a soap bubble is a kind of chain reaction,
i would guess that bubble system may not be able to reach a lower energy state by releasing energy into the enviroment.
so a reaction results in a small energy release but making way for more energy releases as a chain pr... | {
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Finding out Energy value A Lagrangian is given by,
$$L= \left(\frac{\pi}{2}\right)^2 R^d \left[\frac{1}{2}\dot A^2 - V(A_{max})\right]$$
$$E=\left(\frac{\pi}{2}\right)^2R^d V(A_{max}) $$
where V (A) now includes nonlinear terms and E is the energy which is found by taking the appropriate Legendre transform of the Lagra... | The values of $E_{\infty}$ were probably calculated numerically. They explain in the paper that they rewrote the equation for the energy to contain only $A$. So what will have done is input a range of realistic values for $A$ and plot the energy in that range. They found a minimum and calculated the value of $E$ in tha... | {
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Dynamics of controlled overdamped inverted pendulum I wonder how to properly write the motion equations for the inverted pendulum on a cart in case of overdamped dynamics. Imagine the system illustrated in Wikipedia placed in a liquid with high viscosity $\beta$. I completely understand how the system with the fixed pe... | Using $\sum F=m a$ and $\sum M = I \ddot{\theta}$ I arrive at
$$ F = \left( \frac{M+m}{m} \frac{I+m L^2}{L \cos\theta} - m L \cos \theta \right) \ddot{\theta} + \left( \frac{M+m}{m} \frac{\beta \dot\theta}{L \cos\theta} + m L \sin\theta \dot{\theta}^2 \right) $$
now you can play with the values to get what you need.
| {
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Concentrating Sunlight to initiate fusion reaction The idea is to collect sunlight over a large area, and concentrate it down to the nano scale.
For the sake of discussion lets say you concentrate all the light you collect down to 1000 nanometers (1µm).
Questions is... how much surface area sunlight would you need to c... | There is another way to do what you are suggesting. And that is to use an already concentrated form of energy -- lasers.
There are a variety of forms of high energy laser fusion projects underway. Inertial Confinement Fusion is the fancy term for the leading method. See ICF details here I would post some illustratio... | {
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Why is quantum mechanics based on probability theory? What makes us formulate quantum mechanics based on probability theory?
Isn't the real quantum world based on unknown laws to us?
Is it possible that results of an experiment will be measurable in another way but not expected value?
| First question: the origin of quantum mechanics can in certain sense be identified from the intuition of De Broglie that every particle could be described through a wave.
Starting from this point, Schroedinger built the famous equation, to which several interpretation can be given.
Schroedinger himself at first though... | {
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Movie airplanes and suction Having watched a recent action movie (with zombies in it) I wondered whether the suction from a hole in the airplane's hull would really be able to rip out luggage, persons and even seating benches.
To my understanding, "suction" is nothing but the lack of pressure, i.e. the suction in the a... | Mythbusters did the experiment in episode 10. This blog summarises the results.
The experiment was really to see if firing a bullet through the skin of the plane would cause the whole plane to burst like a balloon, and they conclusively proved that this wasn't the case. However the effects of losing a window or of majo... | {
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Cylinder rolling down an inclined plane I've been trying to solve this problem:
A cylinder is rolling down an inclined plane (angle between plane and horizon α). Coefficient of friction is µ. What is the translational and angular speed of the cylinder when it's traveled distance is l( at the beginning v = 0)? Assume th... | Actually you proved, with the third equation, that the translational speed does not depend on the radius $r$ of
the cylinder, or on its mass $m$, but only on $g$ and $\alpha$. Indeed it reads like a differential equation defining the motion of the cylinder.
If the translational speed of the cylinder axis does not depen... | {
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Calculate force of impact on running dog I'm going to hold a dog protection tournament. I'd like to give the audience information about the force of the impact of the running dog when it hit(attacks) the decoy on a huge screen. I'm a software developer student so I have the neccesary skills to development the software/... | The main thing that you need here is the notion of impulse. Consider Newton's second law:
$$\sum{\vec F} = m{\vec a}$$
In our case, the only relevant force is the contact force between our dog and the decoy. Before we move on, let's rephrase this force a little. Acceleration is the rate of change of velocity, and we... | {
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How can an object with zero acceleration move? My physics text has a problem in which it is said that a person moves a block of wood in such a way so that the block moves at a constant velocity. The block, therefore, is in dynamic equilibrium and the vector sum of the forces acting on it is equal to zero:
$\sum{F} = ma... | The forces add to zero, acceleration is zero. If that's how it started, sure, it won't be moving. But initially, someone/something pushed it - forces not balanced, accelerating the block until some desired velocity was reached, and then adjusted forces to add to zero.
| {
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vacuum expectation value and Casimir effect Is it correct to say that the VEV of the SM Higgs is 246 GeV? If so, is the VEV a reflection, or measure, of the Casimir Effect?
|
Is it correct to say that the VEV of the SM Higgs is 246 GeV?
Depending slightly on your conventions then yes, this is correct. If you use the other convention it is $v/\sqrt{2}\approx 174\ \mathrm{GeV}$. Note that the vev is the value of the Higgs field in the vacuum, not the energy! The Higgs field itself has unit... | {
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What is the speed of electrical current in salt water? I am wondering about a specific question regarding the speed at which an electrical current traverses through salt-water / saline.
By this I do not mean the electron drift speed - I mean, at what speed would a current travelling from an anode to a cathode immersed... | The transmission speed of a signal in anything, saline, water or vacuum is given by:
$$ v = \frac{c}{\sqrt{\epsilon}} $$
where $\epsilon$ is the relative permittivity. See this article for a little more info, or Google for many related articles. I couldn't find a figure for saline, but the relative permittivity of wate... | {
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Hydrogen cloud at the universe's beginning? What prevented all of the hydrogen at the universe's start from coalescing into one gigantic star?
| There are two reasons:
First, the expansion of space, which was rapid in the early universe, separated the initial density fluctuations into isolated potential wells. Dark matter and ordinary matter then accumulated into these local potential wells, which eventually become galaxy clusters.
Second, the temperature of th... | {
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Why does the event horizon of a black hole not look like a bright sphere? All infalling matter-energy appears to an external observer as frozen in time at the event horizon. Why then is this horizon not extremely bright due to radiation that is able to escape radially? So should all black holes not appear as a bright s... | Not all black holes have large amounts of infalling matter. A quasar is an example that does, and it does appear extremely bright to an external observer -- often 100 times brighter than an entire galaxy.
| {
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Is it possible to have a geostationary satellite over the poles? My understanding of orbital mechanics is very limited, but as I understand geostationary satellite, they stay in place by having an orbital speed corresponding to the spot they're orbiting over.
So my basic intuition tells me that it's not possible to hav... | As noted you can't be stationary without a really big solar sail or magic rockets. The usual solution to this problem are orbits with long hang times in view of a pole, specifically the Molniya orbit, commonly used by the Russians with a lot of high-latitude territory. Multiple Molniya satellites can provide very goo... | {
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Is it possible for a photon to be at rest? I know it doesn't really make sense if looking at the photon from the wave point of view, but is there any law of physics which prohibits a photon from stopping completely? Thanks.
| The photon is a massless particle. Massless particles move at the speed of light. The special theory of relativity postulates that the speed of light is independent of the frame of reference and the maximal possible speed at which a particle can travel. If there existed a reference in which the photon rested, it would... | {
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Scanning electron microscope imaging In a scanning electron microscopy, secondary electrons are defined as the electrons which obey inelastic scattering whereas backscattering electron follow elastic scattering.
Now my question
Can backscatter electrons produce same level of solution as secondary electrons?
| I don't fully agree with your secondary electron (SE) and back-scattered electron (BSE) definition.
That is right; the secondary electrons are created in an inelastic scattering. However, a BSE can be created in an inelastic scattering, as well. The definition is based on their energy indeed. That is why, in a text boo... | {
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Why does positronium decay into 2 photons more often than into 3 photons? I cannot find the answer to the above question.
I know that para-positronium is created with a probability of $25\%$ and decays into 2 photons, while ortho-positronium is created with a probability of $75\%$ and decays into 3 photons.
I also kn... | Let us start with the wiki article:
The singlet state with antiparallel spins (S = 0, Ms = 0) is known as para-positronium (p-Ps) and denoted 1S0. It has a mean lifetime of 125 picoseconds and decays preferentially into two gamma quanta with energy of 511 keV each (in the center of mass frame). Detection of these phot... | {
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Electrolytes and electric field Let me assume, that I have an arbitrary electric field. Is there any way to determine what happens to this electric field if it is appolied to a charge in let me say water with natrium chloride in it? I guess there is some kind of response to the electric field by the ions, so that it wi... | You want something like the Poisson-Boltzmann equation, or its linearized form, the Debye-Huckel equation.
To illustrate the effect, consider an immobile spherical charge $Q$ at the center of your coordinate system, surrounded by small mobile charge carriers of charge $\pm q$. Gauss's law will give you the potential $\... | {
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First principle calculation of boiling point of water How can we theoretically calculate the boiling point of water at given pressure (other subtle parameters as well, if any)? What is the most accurate (minimum discrepancy with experimental value) computation that can analytically predict the
boiling point of water?
P... | The following article seems relevant.
Simulating Fluid-Phase Equilibria of Water from First Principles. MJ McGrath et al. J. Phys. Chem. A 2006, 110 (2), 640-646. VLab eprint.
Their abstract:
Efficient Monte Carlo algorithms and a mixed-basis set electronic structure program were used to compute from first principle... | {
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Why does a breeze of wind make us feel cooler? In my Astronomy class, I learned that temperature results from the speed of air molecules colliding into your skin. Thus, if the air molecules in the room have a high kinetic energy and thus collide with you at high speeds, your temperature will increase. So how does a bre... | Your body is warmer than the surrounding air and as such when heat escapes from your body it warms up that air; if it didn't, you would have overheated years ago. However, if the air isn't moving, that air around you begins to warm up. Heat transfer is faster if the temperature difference is greater. What wind does ... | {
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Any physical example of an "explosive" differential equation $ y' = ky^2$? I was told that in physics (and in chemistry as well) there are processes that may be described by a differential equation of the form $$ y' = ky^2. $$ That is, the variation of a variable depends from the number of pairs of the elements.
I und... | The population growth of an imaginary species with two sexes, where childbirth and growing up is instantaneous and there's no death.
A dictator decrees that every man and woman must mate, on average, once a year, with each mating producing a child (of random sex). So if there are 1000 men alive today, each woman has to... | {
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Fourier transform of two pulses of light I have laser beam path that fires two pulses of light in a gaussian distribution, so the intensity graph over time is two identical gaussians separated by a distance $t_0$. In other words, a gaussian convolved with two $\delta$ functions equidistant from the origin. When I take ... | As an aside, this effect is called spectral interferometry in optics. But it's even seen in seismology of surface waves that go around the earth many times. If you Fourier transform a long (say 24 hour) surface wave record, the fringes are much narrower than a short recording (say 4 hours). Here is an example of the... | {
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Connection between $\Delta x \Delta p \geq \frac{\hbar}{2}$ and $\Delta E \Delta t \geq \frac{\hbar}{2}$ Is there a way to derive second equation from the first one? I mean is there a connection between those two uncertainty relations?
\begin{align}
\Delta x \Delta p &\geq \frac{\hbar}{2}\\
\Delta E \Delta t &\geq \f... | The uncertainty principle can be seen as a result of space $x$ and momentum $p$ being a Fourier transform pair. The free-particle wave function has, similarly to the exponential $e^{-\frac{i}{\hbar} px}$ an exponential $e^{-\frac{i}{\hbar} E t}$. Thus one could expect a similar uncertainty relation for the variable pai... | {
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Minimum height of mirror required to view image I wanted to know the minimum height of mirror required to be able to view a complete image of a person. I considered the following setup:
$HF$ is the person in question. $H$ denotes the head, $F$ the feet, and $E$, the eyes. For the person to see his complete image, a ra... | To see the complete size of image of a person we must use the mirror of length is equals to half of the length of the person.
If ab is the object then a ray of light moves from 'a' and reflected at p ,then reach at the eye which is at O ,then another ray moves from 'b' reflected at q , then reach at eye O if the ima... | {
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Are there any naturally occurring perfect circles? Given that $\pi$ is the irrational number that occurs with a perfect circle, and perfection is very difficult to achieve through chance or nature, I think that most circles are really ovals, and imperfect.
Are there any proven naturally occurring objects, behaviors,... | I do not know about perfect but naturally occurring yes:
The circles, according to [Norbert Juergens], are water traps created by a sand termite. The termites eat all the grass within a circular patch, exposing underlying sand grains that store any falling rainwater. These barren freckles are works of ecological engi... | {
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What is the meaning of "CW" in LASER? I am reading a user's manual, and the word appears here.
At first, I think "CW" means "center wave".
But later, I find that the meaning of "CW" is "continuous wave".
It makes me confused.
Generally, Laser has a unique frequency. Properly speaking, it has a too small fluctuation ... | Lasers typically operate in two regimes: as CW or continuous wave (the radiation leaving the laser cavity or resonator continuously by a partially reflecting mirror, or as you put it, forming an uninterrupted wave) or as a laser pulse (the radiation leaving the cavity during a short time at a repetition rate, that is, ... | {
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Surface integral of a line Suppose I am calculating the (homogenous) electric flux through a straight line. How do I calculate the surface integral of a straight line? Is it simply a line integral, or is it more complex (I don't think the first is true because if the surface integral is a measure of area, but the line ... |
Suppose I am calculating the (homogenous) electric flux through a straight line. How do I calculate the surface integral of a straight line?
Actually, it's impossible to do this. It's inherent in the definition of flux (in three dimensions, anyway) that you need a surface to serve as the region of integration. You ca... | {
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Can an Incandescent Bulb be considered as a heater? Due to an experiment, I need a small heater (around 70 to 100 watts). I intend to use an incandescent bulb so it can act as a heater. What I wonder here is will a 70 watts Incandescent Bulb be equal to a 70 watts heater?
| From Wiki:
Approximately 90% of the power consumed by an incandescent light bulb
is emitted as heat, rather than as visible light.
So, yes, the incandescent bulb can be used as a heater and, in fact, has been used as a heater. For example, see: Easy-Bake oven.
The original toy used an ordinary incandescent light... | {
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Could an oscillator at a high enough frequency produce light instead of radio waves? Considering that light is in the 400-800 THz range, if you had an electrical oscillator that ran at that frequency connected to an aerial of some sort, would the antenna emit visible light, in the same manner that radio waves emit radi... | In short, no. It's not possible to generate a 600THz electrical signal. The sole reason for this is that electrons could not physically oscillate (vibrate) fast enough using the same methods as when generating radio waves (100MHz.)
One way to have electrons (which make up electricity) have that high of a frequency, is... | {
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The relation between Gauss's law and Coulomb law and why is it important that the electric field decrease proportionally to $\frac{1}{r^{2}}$? My question relates to the third MIT's video lecture about Electricity
and Magnetism, specifically from $21:18-22:00$ : http://youtu.be/XaaP1bWFjDA?t=21m18s
I have watched the d... | The Gauss' law/inverse square connection is explained by Frederic above. But allow me to expand on the fundamental issue.
In a nutshell, if the inverse square law doesn't hold, then the photon must have mass, and hence a finite lifetime. This is explained well in Jackson's Classical Electrodynamics (look in the index... | {
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Heating coffee by yelling? Is it a myth that yelling to a coffee mug will heat it? I have been hearing my friend saying that screaming will heat coffee or water.
| Sound is acoustic energy. Energy is energy. If you yell at your coffee cup, some of the sound energy emitted by you toward the cup will be reflected off, but some will be absorbed into the cup, scattering and becoming generally disorganized into what is known as brownian motion... or in other words, heat. But the amoun... | {
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Projectile Motion Question involving a ball and a ramp inclined at an angle The question is to finde the initial horizontal velocity of the ball at end of the ramp, where it is released.
I know how to do this using gravitational potential energy and kinetic energy ($v=\sqrt{2gh}$), assuming all potential energy is conv... |
Disclaimer: Since this is a homework style problem, I'm not going to write all the details.
Writing down the projectile's equation, and replacing $\theta$ we arrive at the equation:
$$-\frac{x^2}{\frac{4 g (h-y) \left(l^2-(y-h)^2\right)}{l^2}}-\frac{x (y-h)}{\sqrt{l^2-(y-h)^2}}+h=0$$
Solving this for $x$, and simplify... | {
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Virasoro operator in "old covariant quantization" I met some problem about the Virasoro operator in "old covariant quantization" in Polchinski's string theory vol I p 123.
It is given
$$L_0^{\rm m}=\alpha' p^2 + \alpha_{-1} \cdot \alpha_1 + \cdots \tag{4.1.11a} $$
But on p 59,
$$ L_0 = \frac{ \alpha' p^2 }{4} + \sum_... | $(2.7.7)$ is about the closed string, while $(4.1.11a)$ is about the open string.
You can compare the expansions of the closed string and the open string in $(2.7.4)$ and $(2.7.26)$.
You see that the term in front of $-ip^\mu \ln|z|^2$ is $\frac{\alpha'}{2}$ for the closed string, and $\alpha'$ for the open string.
Wh... | {
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The Quantization of Photon Energies Despite Planck's constant being in $E=hf$, it would appear to me that energy is still not discrete, as frequency can be an fraction of a Hertz that one wants. How does this imply that electromagnetic radiation is quantized?
| In my books, the energy content of light is not quantized generally in the same sense that the energy of an electron is not quantized generally. The electron can have a continuous energy spectrum as all other free particles.
It is only particles bound in potentials that get quantized energies, and then the photons that... | {
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Why do we say that irreducible representation of Poincare group represents the one-particle state? Only because
*
*Rep is unitary, so saves positive-definite norm (for possibility density),
*Casimir operators of the group have eigenvalues $m^{2}$ and $m^2s(s + 1)$, so characterizes mass and spin, and
*It is the re... | The irreducible representations of the Poincaré group are labeled by mass $m$ and the spin $s$ [this corresponds to Casimir invariants $m^2$ and $m^2s(s+1))$, so it corresponds naturally to $1$-particle relativist states.
The states corresponding to a representation $m, s$ are labelled $|p,\lambda \rangle$, with $p^2... | {
"language": "en",
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Does $E$ cause $B$ or does $B$ cause $E$ in Maxwell's equations? From the Maxwell's equations we get
$$\frac{\partial E}{\partial x} = -\frac{\partial B}{\partial t}$$
and
$$\frac{\partial B}{\partial x} = -\mu_0\epsilon_0\frac{\partial E}{\partial t}$$
My question is: A change in the electric field causes a chan... | Components of the vectors of electric field and magnetic field are in the same time components of a single Lorentz-covariant tensor of electromagnetic field (http://en.wikipedia.org/wiki/Electromagnetic_tensor ). Separation of electromagnetic field into electric field and magnetic field is not Lorentz-invariant: someth... | {
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How does one prove that Energy = Voltage x Charge? We know
$$E = q V$$
where $E$ is the energy (in Joules), $V$ is the potential difference (in Volts), and $q$ is the charge. Why is this equation true and how we prove it?
| There are various ways to decide which of the assumptions are primary and which of them are their consequences but $E=VQ$ may be most naturally interpreted as the definition of the potential.
The potential energy is a form of energy and the potential (and therefore voltage, when differences are taken) is defined as the... | {
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Why do people use C++ in Computational Physics? Once day, I talked to my Prof. about Computational Physics and he showed me his work. I'm quite surprised that why Prof. use C++ for computation because C++ quite complicated. I heard he talked about memory talk about pointer. Why didn't he use Python or Java that more ea... | I can't really know why your professor used C++, but there are several reasons why you would:
*
*Performance: Scientific computations might require top-notch performance. C++ allows for very low level control over the hardware and has many possibilities for micro-optimization while still providing high-level abstrac... | {
"language": "en",
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How do we know photons have spin 1? Electrons have spin 1/2, and as they are charged, they also have an associated magnetic moment, which can be measured by an electron beam splitting up in an inhomogeneous magnetic field or through the interaction of the electrons's magnetic moment with an external magnetic field in s... | In Richard Beth's Mechanical Detection and Measurement of the Angular Momentum of Light, bright light from a mercury arc lamp was circularly polarized and passed through a half-wave plate (which reverses the sense of circular polarization) attached to a torsion pendulum. A bit of clever experimental design sent the li... | {
"language": "en",
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Why does light diffract only through slits? We can see diffraction of light if we allow light to pass through a slit, but why doesn't diffraction occur if we obstruct light using some other object, say a block? Why are shadows formed? Why doesn't light diffract around the obstruction as it does around the slit?
| Light will diffract around anything, including a block. We simply don't observe this diffraction because most blocks are very large compared to the wavelength of light. One historically-famous example of diffraction around an object is Poisson's spot, in which diffraction around a circular disk is easily observed.
| {
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How does one extract the universal part of entanglement entropy? I want to know how equation 2.11 (page 9) follows from 2.10 (page 8) in this paper.
The two references mentioned just before 2.11 also seem to skip this crucial step. Unless I am missing something obvious everywhere!
It would be great if someone can h... | A taste of answer, skipping numerical coefficients.
The integrand, in the integral$(2.10)$, is $ I(y)= (y^2-1)^{\large \frac{d-3}{2}}$
We are interested at the behaviour as $y \rightarrow +\infty$
So, the integrand is : $I(y)=y^{d-3}(1 - \frac{1}{y^2})^{\large \frac{d-3}{2}}$
So, you get terms, in the integrand, as :... | {
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What is the correct Orientation of Earth's Rotational-axis w.r.t Ecliptic at Equinox? I am boggled with the position of earth relative to the sun at the "Equinoxes".
According to the wikipedia:
an equinox occurs twice a year (around 20 March and 22 September), when the tilt of the Earth's axis is inclined neither awa... | At the equinox the earth rotation axis is in a plane that is
tangential to the planet orbit, and orthogonal to the orbital
plane. Hence it cannot be tilted towards the sun or away from it.
But it does not mean that it is not tilted, but simply that (assuming the axis
has an orientation) it is tilted forward, or backwar... | {
"language": "en",
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The probability of finding the electron in the $\rm H$-atom In the book Arthur Beiser - Concepts of modern physics [page 213] author separates the variables in the polar Schrödinger equation assuming:
$$\psi_{nlm}=R(r)\Phi(\phi)\Theta(\theta)$$
then there a statement that the differential of space in the polar coordina... | $P(r)\,\text dr$ gives you only the probability in an infinitesimal spherical shell around the center. The integration you're expecting is made, when you want to know the probability in a non-infinitesimal shell around the center.
For example, you'd like to know what is the probability of finding an electron between $r... | {
"language": "en",
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Degrees of freedom of the graviton versus classical degrees of freedom I have a puzzle I can not even understand.
A graviton is generally understood in $D$ dimensions as a field with some independent components or degrees of freedom (DOF), from a traceless symmetric tensor minus constraints, we get:
*
*A massless g... | As far as counting d.o.f.s for GR, I believe it goes: Start with a symmetric tensor (10 d.o.f in 4-D). Throw out 4 because of the Bianchi identities (6 d.o.f left). Throw out another 4 because of invariance under space-time diffeomorphisms (in other words, GR is invariant under General Coordinate Transformations, so yo... | {
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No magnetic dipole moment for photon Electrically neutral particles such as neutrinos can have nonvanishing magnetic dipole moments. Spin-1 particles, e.g., deuterium nuclei, can also have dipole moments. Googling seems to show that the Z boson has a magnetic moment.
So is there an elementary argument that explains why... | Here's a possible elementary argument - let me know what you think: if photons had magnetic dipole moments, then photons with parallel-aligned dipole moments would repel and photons with antiparallel-aligned moments would attract, due to their dipole-dipole interactions. This interaction between photons would mess up t... | {
"language": "en",
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"source": "stackexchange",
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Melting and freezing point Why is the melting and freezing point of a substance are always the same? This was quoted in my textbook but they didn't give a reason for this being so.
| Suppose you have a lump of ice, and you want to melt it to water completely, you will heat it.
As the temperature of the ice reaches $0^\circ \mathrm C$, temperature of the ice will stop rising, and all the heat will be used to convert the ice to water. While this is happening the ice and water will simultaneously exis... | {
"language": "en",
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Does a 27 hp engine output the same amount of energy as lifting a 1 ton stone block almost 3 meters per second? I’m trying to get a sense of how much energy a 27 horsepower engine outputs.
27 hp $=$ 20 133 watts (joules/second). Potential energy can be calculated as $E = mgh$ where $g = ~9.8\ m/s^2$ on earth. Therefore... | 1 HP = 745.7 Watts. 1 Watt is a Joule/second, which corresponds to the measure of the rate of energy.
27 HP= 20133 Joules/second. The output of a 27 horse power engine is 20133 Joules per second. The engine of a car is rated a t 50- 120 KW. This engine can make work 672 light bulbs of 30 W. Your analogy is quite right ... | {
"language": "en",
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Does alternating current (AC) require a complete circuit? This popular question about "whether an AC circuit with one end grounded to Earth and the other end grounded to Mars would work (ignoring resistance/inductance of the wire)" was recently asked on the Electronics SE.
(Picture edited from the one in the above lin... | You are correct and your answers are to be found in the characterization of how antennas support a current in and radiate power out of an "RF circuit", they are "RF closed" but not physically closed - similar to capacitors really.
| {
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Is Newton's first law something real or a mathematical formalism? Why do objects always 'tend' to move in straight lines? How come, everytime I see a curved path that an object takes, I can always say that the object tends to move in a straight line over 'small' distances, but as you take into account the curvature of ... | The trajectory of an object you consider is dependent on the scale on which you track its motion. For objects over small speeds or distances, the necessity to correct the trajectory to account the curvature of the earth or the gravity of the moon doesn't exist. However, if you consider longer distances, longer scales o... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/74683",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
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Cosmological models other than FRW The FRW is a nice isotropic and symmetric metric but I think its assumptions are too many. I was wondering about alternative models. Specifically are there any prominent alternatives that have more fleshed out local detail? For example, are there any models that can ascribe a more loc... | There's also the Mixmaster Universe, studied in the 1980s. I'll give a couple references:
*
*Neil Cornish and Janna Levin, "The mixmaster universe: A chaotic Farey tale". Phys.Rev. D55 (1997) 7489–7510. Eprint arXiv:gr-qc/9612066
*Charles Misner's original paper "Mixmaster Universe". Phys. Rev. Lett. 22 (1969) 1071... | {
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How to find the total current supplied to the circuit? Recently, I came across a question based on finding electric current of a circuit. Here's the image...
I know, by using the formula $I=V/R$, we can easily calculate the current as $V$ is given and $R$ can be calculated from the diagram. In the book (from where I g... | You can reduce this to a simple problem by reforming the diagram to appear more meaningful and solvable. First thing to identify is that all points connected by wires without resistance are at same potential since no potential difference is needed for current to flow through them(assumed to have zero resistance). hence... | {
"language": "en",
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accelerated charged particles and interaction with magnetic field In high school we are taught that magnetic field perpendicular to velocity of an charged particle experience perpendicular force that causes it to move in circular path by relation
$$qvB=\frac{mv^2}{r}$$ but in drawbacks of Bohr's theory it was proposed ... | Accelerated charged particles emit electromagnetic radiation. In this case, where the acceleration is caused by a magnetic field and is perpendicular to the velocity, the radiation is called cyclotron radiation. Since the magnetic field doesn't work on (electrically) charged particles, the radius of the charged particl... | {
"language": "en",
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Why don't FCC metals have a brittle-to-ductile temperature transition? I initially thought that it had something to do with the number of slip systems in FCC vs. BCC, but they're both the same.
| For what it's worth (http://materialiaindica.ning.com/forum/topics/ductile-brittle-transition ):
"Let us compare BCC and FCC here. At high temperatures, both of these have mobile dislocations, and thus they can sustain large plastic deformations without undergoing fracture.
At low temperatures however, while dislocatio... | {
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Why does connecting a battery's positive terminal to the negative terminal of another battery not create a short circuit? This is a question regarding the physics behind the observation.
I have guessed the answer to the question, but I may be wrong, so I want to wait for the responses before posting it.
Some major cons... | I think this picture answers the question:
The battery's terminals are not charged. It's just a chemical reaction that starts the charge.
There's a $\text{Zn}$ and a $\text{CuSO}_4$. When they meet each other via a cable, the $\text{Zn}$ gives 2 electrons to $\text{CuSO}_4$ and becomes $\text{Zn}^{2+}$.
$\text{Cu}^{2... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Can I excite an already unstable element for accelerated decay? Say I had an unstable element ready to go through beta decay and I introduced it to high speed electrons: would this lessen the time needed for the product to go through beta decay?
| http://phys.org/news/2011-05-gamma-ray-laser-emit-nuclear.html
http://en.wikipedia.org/wiki/Induced_gamma_emission
Let us not forget Hf-178m2 media snit.
Nuclear decay of any kind must surmount an activation energy barrier or tunnel through it. Pump the barrier directly or wang the nucleus overall and it proceeds at a... | {
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Calculate exact time of Solar midnight I want to calculate real time of midnight (Solar midnight), but I am unable to find any formula or algorithm for that. I have times of sunrises and sunsets for every day of year. How can I get Solar midnight times from them?
Thank you all.
| Maybe finding the Julian date will help you. Julian days are used for astronomical purposes. Every day midnight occurs exactly when the J.D. (Julian date) is *.5. So you can convert the J.D. to U.T. (Universal time) and then to your local time.
| {
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Do bifacial solar panels operate at lower temperature than normal solar panels? Bifacial solar panels have a glass backplane instead of the usual aluminum and are designed to absorb and convert to electricity light incident on either face. Many manufacturers claim that their bifacial panels operate at lower temperatur... | A typical bifacial module will transmit energy that passes through the silicon as well as the energy that passes around the edges of the cells which would otherwise be absorbed in a typical monofacial module design, around 3% of the total. Using the 20% energy transmission value quoted above, a bifacial solar module w... | {
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How do gravitons impact on general relativity? As I'm reading about GR a lot lately, I was wondering: how do gravitons (if they exist ofc) impact the general relativity?
Since in GR, when we look at particles moving in space-time, we are only looking from geometric point of view so to say. Since gravity is represented ... | You would interpret this as gravitational waves being the medium through which changes in matter distributions are communicated to far away places--much in the same way that you see it with retarded potentials in E&M. If you quantize the field, (at this level) all that happens is that this transmission happens in disc... | {
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A naive look at the band of stability suggests a stable isotope of every element. Why is this intuition false? Looking at the band of stability, my first intuition is to conclude (erroneously) that there is a stable isotope of every element that lies close to the belt of stability. Why is this false? (For example, Uran... | Because the Coulomb interaction is proportional to charge, one can say roughly that the energy released in alpha decay is proportional to $Z$, and the energy released in fission is proportional to $Z^2$. This guarantees that sufficiently heavy elements will be unstable with respect to alpha decay and fission.
Any even ... | {
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How do I meaningfully divide by a vector?
How long does it take a baseball with velocity $(30, 20, 25) m/s$ to
travel from location $r_1 = (3, 7,−9) m$ to location $r_2 = (18, 17, 3.5)m$?
I am thinking that it should be the displacement vector divided by velocity. but velocity is a vector and my text is adamant on ... | If at time $t=0$ the baseball is at location $\vec{r}_1$ and moving with speed $\vec{v}_1$ then withough any external influences its position as a function of time is
$$ \vec{r}(t) = \vec{r}_1 + \vec{v}_1 \; t =\vec{r}_2 $$
The above can be made into a scalar equation considering the direction of travel $\vec{e} = \fra... | {
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What is the difference between phase difference and path difference? The path difference is the difference between the distances travelled by two waves meeting at a point. Given the path difference, how does one calculate the phase difference?
| Consider two waves coming from different places and arriving at the same point. Also, let those two waves initially be in phase (i.e when one wave is at a maximum so is the other one). Furthermore, let the wavelength of each wave be the same (i.e. the distance between consecutive maximums).
Now let the first wave tra... | {
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Why does blowing on a candle put it out but sucking doesn't? Alternatively, why does the force created by blowing out air feel so much stronger than the force created by sucking in air?
Ok, so forget the human factor involved in blowing out candles. Consider a vacuum cleaner with a suction end and a blower end. Anyone ... | Watch water going down a drain. It has a rotational symmetry and goes into a vortex, whose boundary conditions perpendicular to the flow cover an area much larger than the hole of the drain.
Watch water coming out of a hose. The boundary conditions defining the vortex are to start with the area of the hose perpendicula... | {
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Does light loses its energy when it passes through denser medium? I know it does not because it emerges out of denser medium at 300,000 KM per second, but according to $E=mc^2$ and given that speed of light decreases inside denser medium with refractive index greater than 1, does not it suggest that energy of light ins... | $E=h \nu$, hence $E$ is proportional to $\nu$,
and $\nu$ does not change when light moves from rare to denser medium. So, it does not effect $E$.
| {
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Central charge in a $d=2$ CFT I've always been confused by this very VERY basic and important fact about two-dimensional CFTs. I hope I can get a satisfactory explanation here. In a classical CFT, the generators of the conformal transformation satisfy the Witt algebra
$$[ \ell_m, \ell_n ] = (m-n)\ell_{m+n}.$$
In the q... | The central charge term as an example of a quantum anomaly; a symmetry that is modified in the quantized version of a classical theory. The central charge is, in fact, often referred to as the conformal anomaly. As di-Francesco et. al. put it at the start of section 5.4.2:
The appearance of the central charge $c$, a... | {
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How to find gain bandwidth with knowledge of source bandwidth and output bandwidth? Deconvolution? Say, I'm amplifying a signal using a device with gain bandwidth, ΔG Hz, which is unknown.
My source signal which is being amplified is known to have a bandwidth of X Hz, and the amplified output signal has a measured band... | I'm assuming that your filter is linear and time invariant, then we know that in the frequency domain
$Y(\omega)=G(\omega)X(\omega)$
and thus conceptually:
$G(\omega)=Y(\omega)/X(\omega)$.
The (first) hard part is that we know that $X$ is band-limited so you'll only be able
to evaluate $G(\omega)$ for those frequenci... | {
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Neutrino power plant Suppose there is a material that interacts with 50% of high-energy neutrinos. To what temperature such material will heat up naturally here on Earth? What power can be extracted from it?
| If we indeed suppose that such an exotic material did indeed exist, then we just have to consider the flux of solar neutrinos from proton-proton fusion. This is about $10^{11} cm^{-2}s^{-1}$ at about 0.4 Mev each.
So a one-metre square sheet of annixxite, would generate, in Watts:
$$
{1\over2} \times 10^{11} \times 10^... | {
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What does imaginary number maps to physically? I am taking undergraduate quantum mechanics currently, and the concept of an imaginary number had always troubled me. I always feel that complex numbers are more of a mathematical convenience, but apparently this is not true, it has occurred in way too many of my classes, ... | It's just a mathematical tool1, which is especially useful to portray some kinds of vectors and sine waves. It doesn't have a real "meaning" here, and we can do fine whthout it (calculations just become more tedious)
For example, in circuit analysis, we convert the coordinate pair $(V,\phi)$ (which isn't linear; adding... | {
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Faster than light in plasma Some plasmas have a refraction index of less than 1. In these plasmas the phase velocity of light can be faster than light-speed. But the phase itself won't transfer information, so no paradox occurs here. But what if I constructed a tube filled with plasma with a torch on one end and a phot... | One way to look at that:
*
*The envelope of the red light pulse travels at the group velocity, less than $c$.
*The fast oscillations within the main pulse travel faster than $c$.
In the end, the torch "lighting up" is given by the group velocity, because the envelope of the pulse is what you receive.
This animation... | {
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Why is the value of thrust for a perfectly spherical event equal to ${\frac{1}{2}}$? In a particle collision the thrust is defined by:
$$ T = max_{\hat{n}} \frac {\sum_i \left| \hat{p}_i . \hat{n} \right|}{\sum_i \left| \hat{p}_i \right|} $$
where $\hat{n}$ is the unit vector that maximizes the ratio of the sums ($\hat... | It seems to me that the convention you're using for thrust is the one used in this lecture, namely that for an event with momenta $\mathbf p_i$, the thrust is $T= \max_{|\mathbf n| = 1} \frac{\sum_i |\mathbf p_i \cdot \mathbf n|}{\sum_i |\mathbf p_i|}$. In the limit of a spherical event, the momenta are uniformly distr... | {
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What is the geometry behind special relativity? Many books on special relativity eventually mention that the geometry of spacetime is special because the metric has a signature $(-,+,+,+)$ which is non-Euclidean. I have encountered many ways this makes it different from normal Euclidean geometry, for example, there is ... | The geometry of special relativity is called Lorentzian geometry, or in full: the "pseudo-Riemannian geometry of Minkowsk spacetime". This is also the Cartan geometry of the Lorentz group inside the Poincaré group.
See on the nLab at Lorentzian geometry for further pointers. See the References there for introductions a... | {
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What is the next step beyond quantum computation? Assuming we develop quantum computers one day, what would be theoretically the next step? Would it be string-theory based computers? How would these computers differ performance-wise (ie what can they possibly do that Quantum Machines cannot)
| You might be interested in this paper, NP-complete problems and Physical Reality, by Scott Aaronson. It doesn't discuss the next step "after quantum computers" per se, but compares the computational power of various physical theories. It surveys newtonian physics, nonrelativistic quantum mechanics, nonlinear correction... | {
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About Pauli-Lubanski 4-vector I saw you answered someone else about the Pauli-Lubanski 4-vector properties. I was doing some simple analysis of polarized leptons, started with the canonical convention of the spin operator wich is just $s=(0,\vec{s})$ and made the simple boost. However I am confused because I have found... | Check the correctness of my expressions, as I'm inconsiderate and could be mistaken.
You only need the expressions for the full momentum tensor:
$$
S^{\mu \nu} = (\hat {\mathbf S} , \hat {\mathbf K}_{1}), \quad \hat {\mathbf K}_{1} = \frac{[\hat {\mathbf p} \times \hat {\mathbf S}]}{p_{0} + m},
$$
$$
L^{\mu \nu} = (\h... | {
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Fabry-Perot cavities and phase difference I'm reading a paper which says
The beam that reflects from a Fabry–Perot cavity is actually the coherent sum of two different beams: the 'promptly reflected beam', which bounces off the first mirror and never enters the cavity; and a 'leakage beam', which is the small part of... | in an F-P cavity such as this...the cavity thickness is quarter wavelength thick, such that 3-5 travels a distance of half a wavelength (180 degrees shift even before accounting for the phase shift due to reflection). when that is accounted for, 1 and 6 are actually in phase. therefore 2 and 6 are still antiphase--anti... | {
"language": "en",
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Power delivered by centripetal force Does centripetal acceleration deliver power to the revolving object?
What I think:
Power is given by F.v Since for a body in circular motion velocity and acceleration are at 90 degrees the power delivered should be zero.
Does changing(increasing or decreasing)centripetal force del... | The answer given by @Danu is correct. This is an elaboration of your case where the centripetal force varies.
Two things can occur when the centripetal force varies:
Case 1:
The tangential velocity varies by an appropriate amount to maintain circular motion.
Basically $$F_{\mathrm{centripetal}}=m\frac{v^2}r$$ should sa... | {
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Proof of conservation of energy? How is it proved to be always true? It's a fundamental principle in Physics based on all of our currents observations of multiple systems in the universe. Is it always true to all systems? Because we haven't tested or observed them all.
Would it be possible to discover/create a system t... | Previous answers such as "all classical equations for energy are constructed to make energy conserved" or using Noether's theorem are correct, but here is another way to look at it.
In quantum mechanics, the energy of an object $\psi$ is defined to be $E(0)=<\psi |H|\psi>$, where $e^{iHt}$ is the transformation that mo... | {
"language": "en",
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How the Lorentz transformation affects the metric tensor? After performing a Lorentz transformation, the orthogonal coordinates will become askew, as in the following figure:
and in such coordinate system, according to this Wikipedia article, the metric will have off-diagonal non-zero elements:
$$g_{ij}=\mathbf{e}_i.\... | In one's rest frame, the x & ct axis are othogonal...x'-ct' are orthogonal to the observer who is moving with the primed frame (rest frame of primed observers)...but w.r.t. one frame the other frame do not have orthogonality...in fact invariance (see Proving that the Minkowski metric tensor is invariant under Lorentz t... | {
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How to derive the two-term approximation for the Boltzmann equation?
Starting with the Boltzmann equation in terms of $f(t,\vec v,\vec x)$ or $f(t,\vec v)$
http://en.wikipedia.org/wiki/Boltzmann_equation
$$\left(\frac{\partial}{\partial t} + \vec{v} \, \nabla_\vec{x} + \frac{1}{m} \vec{F}\cdot \nabla_\vec{v}\right) f(... | G.J.M. Hagelaar and L.C. Pitchford give an elegant derivation of fluid equations in the scope of two-term formulation of the Boltzmann equation. Yours equation above appears in (39) (see "Solving the Boltzmann equation to obtain electron transport coefficients and rate
coefficients for fluid models", http://dx.doi.org/10.... | {
"language": "en",
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What is the universe expanding into? Due to curiosity, it made me wonder what is outside of the universe, is it a new chemical or just empty space?
But empty space have protons appearing and disappearing in it, it basically still has something in it.
(It might have something to do with string theory.)
| The universe isn't expanding "into" anything, it's the space itself that is expanding, or "stretching" if you will! Besides, it's the space between the galaxies that's expanding mostly. Inside them, gravity is overcoming the expansion, so it's really only on the larger scales.
| {
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Plastic bottle floating and sinking I was doing this experiment: i take a plastic cup, put in a container of water, it floats – then when I put a coin in the water, it sinks and rests at the bottom.
So to make the cup sink, i fill it with water – and the cup sinks, but it sinks until it's totally submerged with water ... | this will happen if the cup material's density is very close to that of water itself. completely submerged, the buoyant force on the cup is very similar to its own weight, so it appears to neither sink nor float underwater.
Polystyrene cups are about 1g/cm³. these will do. heavier duty plastic cups will definitely sin... | {
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What limits the mass transfer from a boiler to a condenser in a sealed vessel? If we set up an evacuated vessel where we keep one end at a higher temperature than the other, and then introduce a liquid at the warm end, it will evaporate and condense at the cooler end. If we trap the liquid there, all the liquid will mo... | If the Reynolds number of laminar flow exceeds about 4000, it will go turbulent. Turbulent flow will choke mass transfer, and worse if it a supersonic shock. Rate of heat transfer through source and sink walls is another limit. And if you really juice it (accelerator)...the beta factor in special relativity. That la... | {
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Physical implications behind the exchange antisymmetry condition of fermions Explain the Physical implications behind the exchange antisymmetry condition of fermions. This condition forms the basis of the pauli principle but I can't find/understand what happens physically that requires then the presence of a minus sign... | To perform canonical quantization of a fermion field, we write the field in creation and annihilation operators. Writing the Hamiltonian in those creation and annihilation operators, we find that the energy of a field is unbounded from below (it can be as negative as you like).
This would be a disaster; a field could ... | {
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Why does Joule heating not occur when no current flows through a conductor? Joule heating happens every time when the conduction electrons transfer kinetic energy to the conductor's atoms through collisions, causing these conductor's atoms to increase their kinetic and vibrational energy which manifests as heat. Then, ... | From a classical perspective, "heat" is a measure of the average velocity of a microscopic system. In a conductor at zero potential, all of the particles in the system - electrons and nuclei alike - are moving randomly with no new energy entering the system. An increase in the temperature of the conductor would imply a... | {
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Why doesn't Bernoulli's Principle apply to Current and Resistors in a circuit? Bernoulli's principle makes sense when you apply it to fluids. If you decrease the diameter of a pipe then the velocity of the fluid increases because it needs to keep the same rate of fluid moving through the pipe.
So my question is:
If Vo... | Yes, the analogy is not perfect as voltage can be thought of only as the pressure gradient across the pipe. This analogy will work quite well. For example, as the voltage increases across a wire, the current flow will increase for a given resistance. Similarly, as the pressure gradient increases across the pipe, the fl... | {
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Adiabatic filling of a container Suppose a thermally insulated container is filled with atmospheric air until the pressure reaches 5000 psi. This could represent the filling of a diving cylinder, before thermal dissipation becomes significant.
Initially, then, some mass of air has atmospheric temperature $T_1$ and $p_... | Your calculation looks correct to me (although 5000 psi is a little high for a typical diving cylinder). But this calculation gives the final temperature of the air inside the cylinder. If -- after rapid compression -- we wrap insulation around the cylinder and allow the cylinder and the gas to reach equilibrium, the f... | {
"language": "en",
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Does heat always rise? Does heat always rise in a gravitational field?
I recently read that heat could be traveling down to the deep part of the ocean. Is there some new or old physics that makes heat sink?
| Heat does not rise or sink because it isn't actually a substance, it's energy being transferred. It is hot air which rises. The reason for this is that hotter air is more dilute than colder air. Or equivalently: colder air is denser than hotter air. The temperature of the air only plays a secondary role in determining ... | {
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What is the significance of Lie groups $SO(3)$ and $SU(2)$ to particle physics? I was hoping someone could give an overview as to how the Lie groups $SO(3)$ and $SU(2)$ and their representations can be applied to describe particle physics?
The application of Lie groups and their representations is an enormous field, wi... | I will insert here an experimentalists introduction to SU(2) and SU(3).
Back in the sixties we were organizing the exciting resonance data we got from a multiplicity of experiments into spin Regge poles . Spin was organized in SU(2) multiplets since nuclear physics studies, and the analogue was recognized in isotopic s... | {
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when we rub objects together, what determines which material will pick up electrons? For example We know glass when rubbed by silk will become positively charged while the silk will be charged negative.
What exactly makes glass appropriate for losing electrons in that experiment? (
| There is no single factor that uniquely determines whether a certain material is going to gain or lose electrons, but one of the most important properties is the electron's affinity of a certain material. If the affinity of a material is negative, that material will most probably be a receiver of electrons. If the affi... | {
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Meaning of $C$ in wavefunction equation ($\Psi_{MO} = C_1\phi_A(1s) + C_2\phi_B(1s)$, where $C_1=\pm C_2$) I've just cracked open a biophysics textbook and it's all fine up until the introduction of the letter C in a wavefunction equation, and declaring C1= ±C2
I've had lectures on eigenfunctions etc. before and no rec... | This is the LCAO approximation i.e. approximating the wavefunction of a hydrogen molecule as the sum/different of the two atomic orbitals. If $\phi_A$ is the wavefunction of one hydrogen atom and $\phi_B$ the wavefunction of the other then we guess that the wavefunction of the hydrogen molecule can be approximated as:
... | {
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Can any solid of low heat capacity release or gain energy slower than another solid of higher heat capacity? If our sense of contact temperature is influenced by the thermal conductivity of the material:
Can any solid material with a low heat capacity exist that feels closer to human body temperature than another soli... | Yes. Objects from the freezer that feel unusually cold are good at removing heat from your body (and conversely, they will feel hot if they are put in an oven). When you touch a cold object, the part right near your skin (the "surface" layers) warms up and your skin cools down. It's the cooling of skin that you feel.
I... | {
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On Einstein's equivalence principles There are two foundative Equivalence Principles in General relativity:
Weak Equivalence Principle (WEP): the dynamics of a test particle in a gravitational field is independent of its mass and internal composition.
(WEP is equivalent to say that the ratio between the gravitational m... | EEP has something more than what WEP has. WEP states that in a small reign of space-time,there is no difference for a particle to free fall (to move in a gravitational field) or to move in a box with acceleration $g$ in the inverse direction.
Then Einstein stated more and said that not only for free fall but also for a... | {
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Entropy of the Sun
*
*Is it possible to measure or calculate the total entropy of the Sun?
*Assuming it changes over time, what are its current first and second derivatives w.r.t. time?
*What is our prediction on its asymptotic behavior (barring possible collisions with other bodies)?
| For an ideal gas, you can always relate the entropy to the pressure and density via the relation*
$$
S(r)\simeq P(r)\rho^{-\gamma}(r)
$$
where $\gamma=5/3$ is the adiabatic index. Reasonable (and simple) estimates exist for the density and pressure profiles (e.g., $\rho(r)\simeq\rho_c\left(1-r/R\right)$ where $R$ is ra... | {
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Lorentz Force Law and cycloid motion The question I have is from Intro to Electrodynamics by Griffiths (Page 206, Example 5.2)
http://imgur.com/a/sfBlt
When using the initial conditions $y(0) = z(0) = 0$, I get:
$y(0) = 0$ implies $C_3 = C_1 = 0$
and
$z(0) = 0$ implies $C_4 = C_2 = 0$.
However, how does this work with ... | The general solution you mention is derived from solving a second order linear differential equation. You can differentiate $\ddot z=\omega (\frac EB-\dot y)$ to get $\ddot v_z=-\omega^2 \dot z$ where $v_z=\dot z$. This is a second order differential (linear) equation which has the general solution ( you can try out by... | {
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Notations for statistical / systematic / numeric errors? I constantly see the notation
$$
5.143(13)
$$
for specifying that a value was measures / calculated to be 5.143 with an estimated error of 0.013. I have come to wonder though, just how commonly accepted the notation and variants are. After all, if it is commonly... | In the field I used to work in the notation using ± is quite common. However, the number after the ± is then usually not the standard error but indicates the confidence interval. This is usually 2 times the standard error, but it can also be a factor 3. In you last example this would be 51.3±6.0 using a 'coverage facto... | {
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"answer_id": 0
} |
Why does a laser beam diverge? I was wondering why a laser beam diverges. If all the photons are in the same direction, I would imagine that it would stay that way over a long distance. I am aware that a perfectly collimated beam with no divergence cannot be created due to diffraction, but I am looking for an explanati... | the parallel mirrors cannot be perfectly parallel. they only need to be aligned enough so that photons can bounce between them long enough for lasing to occur. in practice this is not easy, but using intuitive geometry, a shorter and wider (radius) optical cavity allows more tolerance for misaligned mirrors (photons ca... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/79417",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "22",
"answer_count": 7,
"answer_id": 3
} |
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