Simulating Quantum Systems [Split Operator Method]
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- Опубликовано: 4 авг 2018
- More information here: www.algorithm-archive.org/con...
If you want to contribute, here's the github repo: github.com/algorithm-archivis...
The music came from Josh Woodward:
www.joshwoodward.com/
Please feel free to follow me on Twitter:
/ leiosos
Twitch (where I do all the simulations):
/ leioslabs
or Github:
github.com/leios
Also, discord:
/ discord 
Наука
Man, this video took me forever and a lot of stuff happened since my last video. I have a lot of stuff to update you guys on, so my next video will be about that. I will also work on making videos regularly instead of sporadically from now on.
Please let me know what you think!
Obviously, there are a bunch of links in the description with more information!
Great job 👍
Calculus rocks everywhere. It fits perfectly from quantum level to macro level👍
The future of true quantum language mechanics and human-machine hyper linguistics
are on the horizon.
:.:
(interaction; "at the same time...") Non-linear quantum code gives us general time--timing markers
for modeling future time-adjusting neural networks.
;:
(interplication; "flashback to...")
Retrocausality via quantum computation allows a quantum AI
to flash through several quantum enTangeled mode(l)s in a user's
train set, cardinal-ized set timeline of
non-linear yet quantum weighed events; trainSet by the QC AI engineer or
QC-equipped Plant Manager or QC-equipped 3D Print Operator.
:;
(intersequencing; "flashforward to...") The future holds the heaviest quantum weight of Al+l.
;:
You learnt
ereHow
to quantum code/cascade:.
The future.:
This was so good! The simulations have so much potential to explain abstract concepts really clearly. Can’t wait to see more!
Yeah, totally agreed. Sometimes it's nice to just code something up and mess around with it.
Ha... So much potential.
I'm sure this video is great, but it's also about a mile over my head. Maybe I'll rewatch this in a few years
To be fair, I was going to put a disclaimer at the start of the video for that purpose. It is technical, no doubt!
Omg literal mood 😂😂
Great video! This video gave me some great insights into something I was still trying to wrap my head around. I’m excited to see more videos that delve into the actual mathematical intuition for using and generating these algorithms.
I'm glad it was helpful!
This channel is awesome! The idea is awesome! I'm glad I found this.
I'm glad you like the content!
Dude that video was perfect! Thank you, and no you are good no need to haircut :D please continue these videos about simulating quantum world and your methods which corrected my view so much
Hey, great video! I am super excited to hear how you explain your research!
So that's officially 1 vote for hearing about my research.
You have two votes now, LiosOS!
Everyone stop promoting this, it's not that interesting
I disagree chris, I think its very interesting! I think he's using GPU's to play minecraft.
BTW, I miss you everyday Peter....
3rd up here
I've never heard of algorithm-archive. I think I need to go check it out. Great video. I get the feeling you'll have the content I'm looking for. I'm subscribing.
The content takes a while to make, but I am happy with the results so far.
It's ok. From what I gather you're into computer stuff and quantum physics. I like that stuff too. Right now, though, I'm trying to learn calculus so I'll be focusing on that and some other math stuff I'd need to know for quantum mechanics.
Cool video, I'll need split-step Fourier for master's thesis simulations (about em field propagating in an optical fiber) and that's a good intro. Plus, very inspiring.
I'm now hoping to find a video on your channel talking about PhDs, why to for it and the pain of it XD If there's not, please do one!
2:15 Have you seen 3Blue1Brown's video titled "The more general uncertainty principle, beyond quantum"? might give you some of the answers you're looking for.
Yeah, there is a lot of good information there.
Sir! Please share me what software is used in your video? Plz I love it.
This was mainly Julia and GathVL: github.com/algorithm-archivists/GathVL
You have absolutely beautiful eyes!!!
Also, you explained the method really well. I'm lucky to have stumbled across here! Many thanks!
Cool vids ! a channel with much potential......btw I think I seen you watching one of Dan's livestreams?
Hey, thanks for stopping by! Yeah, Dan's a super cool guy. I like his stuff a lot!
In celestial mechanics we split the Hamiltonian this way :
H=H_keplerian_motion + perturbation
Because the two body problem has an analytic solution.
See "high précision symplectic integrator for the solar system" blaines 2013 it should interest you!
Counting in physics: 1, 2, many
You missed an i hbar at 2:30, other than that great video! I enjoyed the visualizations of the simulations
Oops. Shoot. It happens. It should be right in the algorithm archive, though.
I watched a one of your twitch streams!
Woo! I'm glad you found your way here!
I would like to clarify that I found your youtube channel first, then watched your twitch stream later on.
good example of a symplectic integrator
lol at "shrodensher"
Besides that, great video!
mate i literally fucking love you thank you so much this saved me so much time!!!!!
wiggle wiggle wiggle wiggle
Incredible
The really cool part is that splitting the method with 1/2, 1, 1/2 instead of 1,1 make the method of the second order instead of the first. And for a full integration in the whole time domain requires only one step more
Yeah, I didn't cover exactly why this was the case. In fact, I kinda glossed over it in the Archive too. I should go back and add more there. Thanks for the suggestion!
I enjoy the video.
By the way I think it's important to mention that for the imaginary time propagation, the wavefunction must be renormalized after each application of the split operator (SO) with imaginary time otherwise the wavefunction will diminish since the non-real time make the propagation operator deviates from unitarity.
I am interested to know your PhD research topic, in my work I use Crank-Nicholson instead of SO because I deal with strong laser field interaction with atom where velocity gauge is more favorable. Unfortunately, SO method applied to velocity gauge causes commutation error to be too large at shorter distance from the nucleus.
Yeah, this was definitely mentioned in the algorithm archive.
You have two votes now, LiosOS!
Thanks!
Schrödinger UK: /ˈʃrɜːrdɪŋər/, US: /ˈʃroʊ-, ˈʃreɪ-/; German: ˈɛɐ̯viːn ˈʃʁøːdɪŋɐ
This is interesting it's over my head though but I was wondering something about how scales and music are related to mathematics but sound waves aren't carried in the vacuum of outer space. Do similar scales still permeate the universe?
3:33 I just see the tip of a wave which reprsents a dense particle moving whitin an area or range. That could work inr a radar detection system.
What a Man!
Thanks, you too!
30s into the video and I've lost it because of a silly greek letter joke. Well played Sir, you win the internet for today.
"Quantum Mechanics is all about Energy" which is all about Time that virtually exists in the singularity/amplitude of the 1st law, and "moves" (like an Arrow) in the direction of the 2nd law of Thermodynamics?
What advantages does this have over simply computing the change directly, using difference quotients for the derivatives?
Good question! So what you are describing would have a relatively high error. This means you need to use way more timesteps to get a result of similar quality result. In addition, the split-op method can be easily parallelized for work on high-performance computing and such. It also has periodic boundary conditions, which makes it kinda nice to deal with for some other reasons. In many ways, it's also easier to implement this method because it's just a bunch of FFT's and multiplications.
Depending on your situation, there could definitely be better ways to do this. This is just one of the many ways currently used in practice.
One immediate advantage is that every step in the process preserves the normalization of the wavefunction and the total energy of the system. This is because the evolution operators are unitary. By contrast, the difference quotient method involves operators which are NOT unitary.
Great video! I might be missing something obvious, but how can you split up the exponentials like you've done if, in general, V and T do not commute (i.e., e^(T+V) =\= e^T *e^V)? Great video again :)
iirc that's actually why we do the half-step in position space, full step in momentum, and then half step again in position. This allows the commutational error to "cancel" out.
Oh neat, I'll have to look into that. Thanks!
I'm not sure if I am misunderstanding something, but in your visualization... aren't you only seeing one of the basis functions? I've only studied the TISE, but IIRC if you solve the equation for a quadratic well (quantum harmonic oscillator), this probability density is Psi² of the first eigenstate, n=0. Is this correct? If so, how can you get the other states through this method?
The imaginary time evolution will get you to the ground state. Higher order states can be analytically put into the system and then evolved with time; however, there is not a simple way (so far as I know) to find the higher excited states with this method.
LeiosOS Thank you! I guess it makes sense, just wanted to make sure that I wasn't missing anything. I'm hoping you follow up with another video on quantum mechanics and computational methods, it's pretty interesting.
no. psi is not always correspond probability of detection. in Dirac equation psi correspond to the charge density ☺
To be fair, psi is not |psi|^2. I thought that the charge density was always related to |psi|^2, not psi?
x -> |x|^2 is correspondence
elation, is not it?
I don't think the split operator method is the best method for the Schrodinger equation. You can do a Crank-Nicholson timestep in O(n) time, but the fourier transform is O(n log n). And if you *really* want to be efficient, you should go with Multigrid or a Krylov subspace algorithm. There is a whole field of applied math working on developing efficient algorithms to solve PDEs, and they've build a lot of cool stuff!
That said, this is a really good introduction to these algorithms, which are a good place to start.
I completely agree. There are many other methods for solving the Schrodinger equation, but the split-op method is super easy to implement for an n-dimensional system and is used all the time in research for this purpose. It's got a relatively low error too, so it's not bad.
I know our lab uses split-op for everything. I use it because it's easy to parallelize on a GPU. Everyone else uses it because it's super easy to code and doesn't require deeper knowledge of diff. eq.
Crank-Nicholson is on the list of future algorithms to cover, but it takes some time to build into because it requires Backward Euler / implicit methods. Diagonalization algorithms are also on the list, but again... it will take a little while to tell the whole story of those algorithms (if that makes sense).
Oh yeah! I just remembered writing Schrodinger solvers and waiting several days for them to finish, and then taking a class in the applied math department, and realizing that I could cut it down to hours or even minutes.
KSP algorithms are super complicated, but you don't have to write them 😃. PETSc and SLEPc are libraries that do most of the heavy lifting.
Yeah, again agreed. That said, it's always nice when you can code things up without relying on external libraries to do it for you. I know no matter what you do, you will not be able to write code that is as quick or robust as certain libraries, but I cannot help trying at least once to learn how the algorithms work underneath.
I suppose I can just do a full step in position space if my potential is static? I promise to start and to finish with half steps.
and is this "Quantum Systems" have anything to do with time travel wise and quantum speed, like spaceship stuff?
wazzup shaggy 2.0!
so wait, this is basically leapfrog integration but for quantum mechanics right?
I find it very hard to click the like-button more than once... an unfortunate random restriction.
I'm glad you liked the video! One like is more than enough! =)
I'm curious: what software do you use to animate these things?
This was done in a mixture of julia and GathVL: github.com/algorithm-archivists/GathVL
I have a doubt regarding the Split-Operator Method implementation presented at the Algorithm Archive (www.algorithm-archive.org/split-operator_method/split-operator_method.html), specifically the Python implementation. To whom should I address this doubt?
Please create an issue on github. Sorry, this was caught by spam for some reason.
I am feeling sleepy now 💤
I'm sorry the video bored you!
Could I ask you a humble question: Are you studying Physics or Computer science ? Is it bachelor or master level ?
I do something in-between physics and computer science. PhD student.
What is that field called ? Large scale simulations, Physics of computation, Efficient algorithms for simulations, etc. ?
Right now, I guess you would call it computational physics; however, my exact research is in simulating quantum systems on massively parallel GPU devices.
Thanks.
4:16 where did the two 2s go?
So, I screwed up. Every time I wrote the Schordinger equation, I forgot the squares on the momentum-space operator, so I added them in in editing. This meant that when I popped up on screen, the 2's had to go away. This happened for both times I wrote down the Schrodinger equation, but was only really obvious the second time.
I knew someone would notice them!
I used to forget the - sign in my early days until I realised that the - comes from i^2. Once I made the connection from p = -i h d_x to p^2 = - h^2 (d_x)^2 I never forgot it again.
ha .. went back to see the first time
Yeah, I am just used to writing it a different way, so I forgot when writing my notes to put on the blackboard. It happens.
"where did the *two 2s* go?" - The ballet?
Omg are you the boy who made that viral video about the 4th dimension??? Omg you are so similar
Yes... I didn't realize it was viral, though...
I can confirm he is the boy that made the fourth dimension video and I sit next to him everyday. He's depressing
I sit two desks away from him and even at that distance his charisma is intoxicating.
@@LeiosLabs | Not like VIRAL but it was very well done! How time flies by! You really changed a lot!! Jeez (in the good way)
@@mossylikescake | i am not understanding your comments lololol (him who)
For your information, Schrödinger is pronounced with a hard g.
Fair enough. I work in an international quantum lab, so I hear it pronounced a bunch of different ways.
Schrödinger is neither pronounced with a hard g nor with a soft g. The g is only there to change the sound of the n, but isn't pronounced seperately at all.
German is not my primary language (nor is English), so I would listen to ehtuanK :) To my ear, it is close to the sound in "singer". Would you agree?
yes. One could also just look up the pronounciation on his wikipedia page: [ˈʃʁøːdɪŋɐ] (or /ˈʃʁøːdɪŋər/ in broad transcription)
Where can I go to learn more about imaginary time?
I'll put more info in the algorithm archive soon. It's on the list of things to talk more about. That said, it's basically a mathematical construct to make the math work out for this example.
Umm... Are you Farkle Minkus?
Regardless, this was a very good explanation.
No, sorry. I'm James. Nice to meet you!
for myself 1:39
wi don't even know how I ended up here.
Soooo...why is the function so wiggly ? =P
What's your profession? Are you a quantum computing guy or something?
Who is this guy?
When you say imaginary do you mean complex numbers?
Yeah. Complex numbers.
If you need me to implement any of the algorithms in Python, hit me up :D just as long as there's at least sudo code lol
*pseudo
+TAP7a I spend too much time on Linux lol
I spell it wrong every single time...
We actually already have a python implementation (someone did it before the video came out)
The convolution algorithm doesn't have Python yet
so, what exactly is this in simple English?
It's like fractals.. you have to multiply by i..
Hahaha.. simultaneity be damned..
Without the I each iteration you are just playing with numbers... the i gives each iteration the randomness we need to study quantum effects
My friend Peace the phone doesn't know this stuff yet... he is giving capital I... instead of the i of our imagination
I am working on exactly this to go back and look at quantum electrodynamics where Feynman had to... re normalize his math to get rid of those pesky infinities... the way we look at particles in the Universe has everything to do with that.. he called them spheres... now.. if anyone dares to say it might be different, unless you are Leonard Susskind, people say .. yer a nut, boyo... it's the reason he had to not just normalize his statistics to get a balanced view of his data.. he re normalized... because the infinities wouldn't go away... he said he thought it might be cheating ... with a laugh... it was not so much cheating as forcing the math to present the Universe the way he wanted... it is a recurring theme with him... particles are much more complicated.. and if they were really spheres stuff wouldn't do what it does
But they are not spheres.. particles are stuff... we say they have "mass".. and mass and energy are equivalent to each other by a function of the square of the speed of light ... great deal of energy stored in one very small locus... how do they do that?.. therein lies the answer
Boring. Where are the speedos?
On my second channel! How many times do I have to say that?