Ah. Finally a topic I know something about !
There are many places in physics where the YB equation pops up. I can think of two at the moment.
a. Exactly solvable lattice models
b. Quantum Computation (QC)
It is the second application I find most exciting, so I'll focus on it.
The canonical ...
Sometime ago, I took part in MIT's EFT mooc. I did not go through the whole thing, and the level was also a bit above me. Are there any EFT tutorials on blogs or a quick short text written for hopeless people like me? I understand some of the concepts like matching, at least for simple quantum mechanics scattering problems. I am looking for something that has the full qft considerations, and eft concepsts introduced for a very simple case. What is the best text for this?
@vzn Severe financial circumstances. I have been off for close to 4 years now. In that time I sort of picked up Physics, Math and CS while hanging around grad schools lol. I like HEP, Condensed Matter and Quantum information. I am not sure what the best options are, I have some interests in a few particular topics but I am super flexible.
@kevinTahN. so a "near miss". know another close cohort who dropped out of EE with 2yrs to go & then went back, wild story. it can be done. strongly advised him against leaving at the time tho :|
Hi @JohnRennie I'm afraid I can't discuss the question at the moment - I have a lesson starting soon - but could if possible in around 5 hours time? (7:20 for me at the moment, and I have a free at 12 - I'm guessing time difference could be an obstacle here)
@skillpatrol No, though I was at Cambridge when he was the Lucasian professor and I did occasionally see him being pushed along in his wheelchair usually by his students.
It seemed rather gauche to go running up to him and ask "hey are you Stephen Hawking or just some other dude in a wheelchair" :-)
@skillpatrol : yes. Because of stuff like this: "Hawking explained that M-theory allows the existence of a “multiverse” of different universes, each with different values of the physical constants. We exist in our universe not by the grace of God, according to Hawking, but simply because the physics in this particular universe is just right for stars, planets and humans to form.
There is just one tiny problem with all this – there is currently little experimental evidence to back up M-theory. In other words, a leading scientist is making a sweeping public statement on the existence of God based on his faith in an unsubstantiated theory."
@yuggib : Newton didn't go in for the speculative stuff that cannot be disproven. That's all Hawking has ever done.
If you simplify the expression with the logs, assuming that $x$ is real, it just works out to $\theta(x - a) - \theta(x - b)$ - it's a rectangular window function
With zero second derivative in $x$, there is nothing causing the solution to change over time
It's like one of those 1=2 proofs where you wind up dividing by zero without realizing it
Oh, so like the air (or whatever) all starts in a certain region $[a,b]$ with uniform density and then at $t=0$ you remove the barriers keeping it there and allow it to diffuse away?
Ah, now I think I get it
It has to be an issue with the boundaries, $a$ and $b$. I bet when you use Mathematica to do one of these Fourier transforms (or inverse) it ignores the points at $a$ and $b$.
Just plug it into DSolve and you get $$c(x, t) = \frac{1}{2}\biggl[\operatorname{erfc}\biggl(\frac{a - x}{2\sqrt{D t}}\biggr) - \operatorname{erfc}\biggl(\frac{b - x}{2\sqrt{D t}}\biggr)\biggr]$$
note that $\operatorname{erfc}(z) = 1 - \operatorname{erf}(z)$
Well, anyway... it's beyond my abilities at the moment to figure out exactly why the Fourier transform method fails, but it definitely has something to do with those boundary points $a$ and $b$.
That would be a pretty good question to write up and post on either Physics or Mathematics. My guess is it better suits the latter.
@user507974 I think @DavidZ is exactly right - the lack of derivatives means your equation can't drive the evolution of the rectangle. It's a "meta-stable" configuration, you only get evolution once something disturbs that configuration and creates a gradient that then can drive the diffusion
Easiest way to understand Hawking radiation I'd say is that around a black hole, there's an outward flow of positive energy and inward flow of negative energy
@JohnRennie The vacuum is empty, there's no cancellation going on, it's the ground state. The positive modes become particles, the negative modes become antiparticles. A state with positive and negative modes in it is very different from the vacuum, it's rather explosive!
@NoahP because the state that corresponds to a vacuum for someone falling freely into the black hole corresponds to an excited state for we observers far from the event horizon.
lol, my Mathematica notebook has like 5 lines of integrating Bessel functions (what I'm supposed to be working on) and then pages of this stuff about the diffusion equation
@NoahP although ACuriousMind smiles, he's quite right. This is a fundamentally horrifically complicated area. At school level all we can really say is that in curved spacetime the quantum state isn't uniquely defined so depending on where you are you will see what appears to be a vacuum radiating.
But the point of all this is that it nowhere involves pairs of virtual particles.
@JohnRennie @Slereah @ACuriousMind @Danu If I were to do some work on refining this and then get back to you later with a new understanding, could you give me some constructive criticism (Unlikely to be 100% praise...) on it?
@NoahP of course, though I fear we haven't given you enough for a good description. I'll have a scout around and see if I can find some reasonably accessible explanation.
@user507974 The "right" way is to note they pair up as 1+100+2+99+3+98+4+97+--- = 50*101, the wrong way is to waste your time by successively adding 1+2+3+4+...
@Slereah i find that a surprising number of tall tales are true, the weird things that will happen around me that constitute the realm of things you should not see happening is very high
most of them are little unnoticeable details that are not very likely but just occur
for example, as I was biking home I happened to randomly think about the movie She's the Man, and when I arrived home after I took a short nap there was a post related to that movie on a (distant) friends timeline
I've had advertisements that directly targetted things I did that have never been done online and sometimes there are very weird sale patterns around me
newegg happens to put an item on sale the day after I search up on it, about 4 times (and its not like they have a massive inventory)
I heard Gauss's primary school teacher gave some busy-work to his class: to add all the numbers between 1 and 100 up. Gauss immediately wrote 5050. His teacher was shocked, so she told him to add up all the numbers to 1000. And just as quickly he wrote 500500.
Did Gauss derive the $1+2+3+\ldots+...
@skillpatrol : stuff that related to scientific evidence. I wouldn't turn my back on the maths. It's just that I'd be talking about electrons and gravity instead of branes.
@yuggib : I agree it isn't a distinguishing feature of mathematics. As for Hawking not being a mathematician, let's just say I'm not a Hawking fan. Nuff said.
\includegraphics cannot handle filenames that contain more than the one dot, separating the filename from the extension. Apparently it uses everything after the first dot as extension and then, of course, complains about an unknown graphics extension.
This is annoying as I very often have filena...
@skillpatrol statistically i suppose its possible for that to occur I suppose but I dont like to believe I was looking off to the side and missed the chance to observe macroscopic tunneling
@skillpatrol mine will probably just be like an 80 year cycle
i remember i opened a notebook on the first day of university, when i was still waiting for fin aid to come, and in that notebook (I'd bought like 3 years ago), there was birthday card with $100 in it
its really fun to be me though with all this shenanigans
Ah. Finally a topic I know something about !
There are many places in physics where the YB equation pops up. I can think of two at the moment.
a. Exactly solvable lattice models
b. Quantum Computation (QC)
It is the second application I find most exciting, so I'll focus on it.
The canonical ...
