I'm reading von Neumann's book on QM and I'm slightly confused by a simple point. He writes, "The energy is a given function of the coordinates and their time derivatives: $E = L(q_1,..,q_k;\dot{q}_1,...,\dot{q}_k) = H(q_1,...,q_k;p_1,...,p_k)$. (This $H$ is the Hamiltonian function.)
Does $L(....)$ denote the Lagrangian? Why does the second equality hold?
I haven't changed the H2O NaOH mixture for about a month and a half, should I make a new one or that's not the problem? I have another capsule where it looks like milk, I'll try with it.
Also when I apply the electric field on the milk-like mixture, the I should see a clear distinction between the TiO2 particles and the water (they should split)?
Visual Studio 2019 now supports Python, and it seems to work very well. I've only had a quick play so far, but everything I've tried works fine including packages like numpy that aren't included in a default Python install.
One thing I would actually like to know is why there is that general notion that you can only do Hamiltonian mechanics on a globally hyperbolic spacetime
You can define Hamiltonians on arbitrary spacetimes
I'm guessing you can't have unique hamiltonian evolutions, though
But nobody ever bothers to prove it
I have even see people defining Hamiltonians on non-globally hyperbolic spacetimes bc why not
So $S = \int L dt$ is the action, and since $L = p \frac{dq}{dt} - H$ is the Legendre transform of $L$, we see $H = p \frac{dq}{dt} - L$, so that $S = \int L dt = \int [p \frac{dq}{dt} - H]dt = \int (p dq - H dt)$. From this you can see $\frac{\partial S}{\partial t} = - H$ easily.
Now the Schrodinger equation in quantum mechanics comes from the fact that a wave function $\Psi$ should reduce to a function of the form $\Psi \approx e^{i\frac{S}{\hbar}}$ in the classical limit, where $\hbar$ is a quantity that makes the argument of the exponential function dimensionless, and so taking the time derivative
$\frac{\partial \Psi}{\partial t} \approx \frac{\partial }{\partial t} e^{i\frac{S}{\hbar}} = i\frac{1}{\hbar} \frac{\partial S}{\partial t} e^{i\frac{S}{\hbar}} = - \frac{i}{\hbar} H \Psi $ we find the Schrodinger equation $ i \hbar \frac{\partial \Psi}{\partial t} = H \Psi$
I came upon the concept of irreducible vector-spinors while trying to simplify an expression involving the gravitino field.
It is claimed that an irredicible vector-spinor is gamma-traceless, i.e.
$$ \Gamma^M \psi_{\alpha M} = 0.$$
Are there conditions on the irreducibility besides the above eq...
I read this Under what conditions is a vector-spinor gamma trace free. And also read many papers about higher spin, but no one explains why irreducible spinor is gamma traceless spinor? Can anyone explain this?
To solve Schrodinger's equation for the hydrogen atom (time-independent), $$ \dfrac{-\hbar^2}{2m} \nabla_x^2 \psi(x) + v(x)\psi(x) = E \psi(x) $$, How to convert the nabla from the cartesian coordinate to the Spherical ones?
@AbhasKumarSinha I'm not smart enough to understand those complicated equations and therefore I cannot have those issues. (of having problems with them (the equations))
@PM2Ring JohnRennie recommended I try with water first, see if it works, and then move to oil and so on. I tried with baby oil, but I can't charge the particles. If I put a little water in the oil with TiO2 particles, a bubble of water and TiO2 forms. The drying method doesn't work since removing the water removes the charge.
You also need a surfactant. A little bit of detergent added to the TiO2 sludge while it's evaporating will probably work. John Rennie may have better suggestions.
@NovaliumCompany Hopefully, a little bit of water will be ok. That way, you have your charged particles, some surfactant, and a liquid that's mostly non-conductive.
As I said a few weeks ago, your cell is basically a capacitor, so you want an insulator between a pair of conductive plates.
So here's what I can try: I put a solution of H2O, NaOH and TiO2 particles to evaporate on a plate while I put some soap. I don't want the whole water to evaporate so I leave a bit. I end up with a sludge. I put that sludge in baby oil.
@AbhasKumarSinha It's the material that Amazon Kindle screens are made from and stuff. It's "electronic paper". You can think of it kinda like paper where what is written on it is electronically controlled, or like a screen that doesn't use any backlighting, and only changes the colour of reflected light
Traditional soap is a sodium (or potassium) salt of a fatty acid. That makes one end of the molecule oil-soluble, and the other end water-soluble. But I'm not sure if that sodium will be a good thing or a bad thing in the e-paper cell.
@AbhasKumarSinha Well, it probably can, and I'm pretty sure that's what Novalium is working towards. Step one is getting a single "unit" which he is able to flip between two colours using electricity. I get the feeling that Novalium is trying to do it for themselves anyways, as a learning experience.
@AbhasKumarSinha I think the point is that it's difficult to extend it to more before you even have one working. Typical e-paper is basically a normal screen; but making it from scratch with your own plans would require you to make sure you can get a single pixel to work before trying a screen with multiple
#Chandrayaan2 is now on it's mission to the moon, will be touching the lunar surface tomorrow at 1:30 AM IST, I tell you, if it gets successful, then I won't sleep till 2:30 tomorrow.
As distance is the second integral of acceleration and $F=ma$, the answer is no because there are no forces that increase with distance and especially none that diverge for any non-zero distances.
Ah, I see where you're coming from. The poorly understood reason is called "dark energy", and it involves general relativity where we shouldn't apply Newtonian physics as there isn't really any "force" to the expansion. See physics.stackexchange.com/q/2110/50583 and its answers for some discussions of the universe's expansion.
@Ultradark Well, the expansion is mostly linear, with a small acceleration component, which can often be ignored, unless you're looking at truly vast distances & time scales. And even when you don"t ignore the acceleration, the expansion rate is so small that gravity overwhelms it, until you get to the scale of galaxy clusters. Individual galaxies within a cluster aren't expanding away from each other because they are bound by gravity.
Following a suggestion from this thread on Computational Science, I had a brief look at a plot of
$$
f(E) = \frac{U(-E,1,R)}{\Gamma(1+E)},
$$
i.e. the Tricomi confluent hypergeometric function as a function of its first parameter, which looks like this
for $R=10$. I'm struck by the resemblanc...
@EmilioPisanty I see what you mean, yeah. I’m not yet seeing an easy relation tho
It may be that, within that regime, a similar approximation applies to both
Which would not be so terribly strange: the Airy and Tricomi functions are both obtained as particular confluences of the hypergeometric function
So it may be that they have a nice relation in that regime even tho they don’t everywhere
On those grounds, it may be worth comparing their contour integral representations since that’s what I’d subject (after sensibly deforming the contour) with Laplace’s method
That said, there is one obvious difference between them: You’re plotting U/Gamma as a function of its parameter, whereas Ai is plotted with respect to its argument (since it has no parameters)
So that may complicate any analogy between them in this regime
I just ran into something interesting. Humans can live near a black hole. The dead zone in the 80'S Sagittarius A has a dead zone of 3,200 light years. No wonder why black holes are only known as deadly. But a recent experiment has proven quite the opposite. According to Harvard University Manaswi Lingam the dead zone is much smaller than originally thought at 140 light years.
At 140 light years from the black hole Sagitarious A life can begin and flourish.
At that distance the habitable zone begins. At this distance planets the mass of the earths will have there atmospheres. The radiation at this distance isn't so strong as to destroy all living things. The ultraviolet radiation at that distance can breakdown molecules to produce proteins, lipids and DNA. The radiation can also stimulate photosynthesis. Which is the most important processes for oxygen.
1 Million earth mass planets can be in the habitable zone. This is according to the astrophysicists at the Bordeaux observatory. They can be placed on 400 rings with 2,500 planets on each. The distance from each planet will be 10 times less than between the moon and the earth.
@rob Not at the moment, but I can get it. I hate timing though. I have to go do some stuff. But I'll get that paper for you at least by tonight. I have plans to meet up with some students of mine. I'm teaching a HV electricians course. I'm super young, but I have a lot of skills. I am of course going to be supervised by a few other HV Electricians.
Following a suggestion from this thread on Computational Science, I had a brief look at a plot of
$$
f(E) = \frac{U(-E,1,R)}{\Gamma(1+E)},
$$
i.e. the Tricomi confluent hypergeometric function as a function of its first parameter, which looks like this
for $R=10$. I'm struck by the resemblanc...
