Before You Start On Quantum Mechanics, Learn This
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- Опубликовано: 15 дек 2021
- Quantum mechanics is mysterious---but not as mysterious as it has to be. Most quantum equations have close parallels in classical mechanics, where quantum commutators are replaced by Poisson brackets. Get the notes for free here: courses.physicswithelliot.com...
You can't derive quantum mechanics from classical laws like F = ma, but there are close parallels between many classical and quantum equations. Many fundamental quantum equations are expressed as a commutator of operators, such as the canonical commutation relation and the Heisenberg equation of motion. These equations have classical parallels where the quantum commutator is replaced by a classical operation called the Poisson bracket, up to a factor of i hbar. I'll show how Poisson brackets work, and how they mirror these key quantum equations.
Get all the links here: www.physicswithelliot.com/poi...
Intro to Lagrangian and Hamiltonian mechanics: • Lagrangian and Hamilto...
Introduction to the principle of least action: • Explaining the Princip...
Noether's theorem: • Symmetries & Conservat...
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About physics mini lessons:
In these intermediate-level physics lessons, I'll try to give you a self-contained introduction to some fascinating physics topics. If you're just getting started on your physics journey, you might not understand every single detail in every video---that's totally fine! What I'm really hoping is that you'll be inspired to go off and keep learning more on your own.
About me:
I’m Dr. Elliot Schneider. I love physics, and I want to help others learn (and learn to love) physics, too. Whether you’re a beginner just starting out with your physics studies, a more advanced student, or a lifelong learner, I hope you’ll find resources here that enable you to deepen your understanding of the laws of nature. For more cool physics stuff, visit me at www.physicswithelliot.com. 
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Again very good. Yes I would appreciate to see Noethers Theorem worked out in Hamiltonian formalism.
Me too!
Thanks Bart!
I'd like to see it too
I'd like too... : )
Yes please thank you. Your videos are so simple to follow, thank you so much.
Your videos are crystal-clear, beautifully laid out and follow a precise progression, treating not so easy topics that are usually not well understood or explained with a lot of confusion.
I think your videos are among the very top quality materials on physics divulgation, and I'm sure that more and more people will join. Keep up the excellent work!
So glad you’re enjoying them! Thank you for the kind words!
@@PhysicswithElliot bro changed the board to black board it looks ugly
@@masternobody1896 I like the blue board
I like the blue board
que Harold and the Purple Crayon on magnificent 8 mm in 1st grade!
I remember finding this connection between QM and CM really intriguing when I first learned it, but I'm also a bit sad that I've never learned a deeper reason for why the "replace Poisson brackets with commutators" rule makes sense. Do you know of any deeper algebraic or physical reason why this connection exists?
Hi Chris, thanks again for sharing the video! The basic reason is that in quantum mechanics we want to represent classical functions like position and momentum as operators acting on the space of states. And we want that representation to respect the Poisson bracket structure that we started with, meaning that the commutator of operators satisfies the same relations as the Poisson bracket of the corresponding functions. But they can't literally be equal, because [\hat{x}, \hat{p}] has units of kg m^2/s, so we need a factor of \hbar to get the units right. And we want \hat{x} and \hat{p} to be Hermitian so that their eigenvalues are real, because those are the numbers we measure. But the commutator of two Hermitian operators is anti-Hermitian, so if it's going to be a constant it had better be a pure imaginary one, so we also need a factor of i. That gets us to [\hat{x}, \hat{p}] = i \hbar. There's other ways of getting at it, but hopefully that helps
If u watch Lenny susskinds lectures on quantum mechanics he gives an explanation that’s quite satisfying I forget what lecture tho maybe 4-5.
@@PhysicswithElliot Thanks. I'm curious what "other ways" there are. You can feel free to share links to other sources if it will save you time.
Just as Hamilton's equation dx/dt = {x, H} determines how x changes with time, the Poisson bracket {x, p} determines an action of p on x by dx/da = {x, p} = 1. The solution is just x = x0 + a, where a is some arbitrary parameter, so that the action of p on x just shifts it over. We say that momentum is the generator of translations.
In the quantum version, we look for a corresponding unitary operator U(a) that shifts the position operator by a constant: U^{-1}(a) x U(a) = x + a. Since U is unitary, we can write it as U(a) = e^{-i a p/\hbar} for some Hermitian operator p. Then at leading order in a this equation is (1 + i a p / \hbar)x(1 - i a p / \hbar) = x + a. The LHS is x - i a [x, p] / \hbar. Therefore [x, p] = i \hbar. This is the quantum version of the statement that p is the generator of translations in x.
@@PhysicswithElliot Thanks. Strangely enough I was familiar with the quantum version of "momentum is the generator of translations" but not the classical version.
i love how your able to stay focused to the topic at hand to avoiding long tangents. It makes the videos so much easier to digest.
The quality and level of the videos is just too good and provide much insight.
They seduce you to pick up a pen and paper to do the calculations yourself.
I could not resist and am proud to have become one of your patreons just now.
Keep up the good work!
Thank you so much Bart!! Much appreciated!
Really nice and clear account of link between poisson brackets and commutators - thanks for explaining this Elliot!
Men this videos are a jewel. You have a knack for explaining physics and it's clear that you put the effort into understanding the concepts and a lot of effort into this videos. Thanks for sharing. I hope the channel grows.
I just started learning about QM and your videos open my mind in a very weird and beautiful way. Thank you so much!!!
As I'm going through my physics degree, this video is helping me a lot to better understand my classical dynamics course, thank you very much for these videos, I will follow up on your very useful video uploads.
this is a really good and important place to start. especially for those who are interested in the theory since these concepts are also the cornerstone of quantum field theory. you could follow this up with a video about the interaction picture and other pictures
From Vietnam with love, thank you so much for clear and easy-to-understand video.
My classical mechanics course glossed over hamiltonian mechanics, but your video was still very clear. Gonna go read up on hamiltonian mechanics now
thanks, getting so much closer to understanding the Quantum world !
This was so well explained that I subscribed. I'll have to check out some more of your videos.
Yes Elliot, definitely interested in Noether's theorem and its applications.
How nicely put. 👍 Thanks Elliot.
This is where the concept of spin (a form of angular momentum) gets weird as it has no classical counterpart that you can relate it to😍. I love physics.
As someone who is going to take QM in my next semester, thanks for the help! Thankfully we covered a lot of what you said in CM, but the video is a cool summary and refresher on the topic. Especially necessary when you have so many other subjects too.
Glad it helped!
these are some the best physics videos i’ve ever seen! please keep doing what you’re doing
Thanks Kashu!
Fantastic. Please make more videos like this one.
love your work, im currently studying quantum / physical chemistry and your work helps me a lot to understand certaint topics. thanks for you videos, love from Slovakia
Love to hear it, Adam!
I lovee this!! Thanks for always making us learn something new clearly!
Thanks Aman!
Thank you very much! I highly appreciate your videos. Please, show more content of QFT.
Really a good youtube on this subject - very important concepts; a very important topic that deserves more detail and examples in a follow up vid, please!
Thanks Dennis!
PLEASE EXPLAIN. You have no idea how much researching I have done just to understand quantum mechanics. You are an absolute genius science educator. Keep up the good work man!!!!
Glad it was helpful!
Definitely interested in the Hamiltonian version of Noether's theorem.
Once again thank you for a lucid and engaging presentation .
Thanks Robert!
Thank you so much sir for this video ❤✨
Gold.
Awesome video, keep it up!
Thank you so much! Subscribed.
Thanks Daria!
Released at the perfect time! Wonderful explanation, kudos to you Elliot, thank you!
Also could you do a video on Lagrangian and Hamiltonian dynamics. That topic also looks very confusing but after watching this video, I discovered what complicates topics are the lack of emphasis given to the foundation! Thanks again
No way!! you already have it! :) Sir, you are something else. Man like Elliot, legend!
Glad it helped Abdullah!
Yes, please do the more advanced explanation. If you have any experience with Bessel functions, I would like to see something on that. Really enjoy these videos!
Thanks Eric!
This was awesome little insight!
Glad it was helpful Cesar!
Thank you!
very helpful on such an abstract topic.
Thanks Aihara!
This is really amazing,
Watching your videos is really beneficial for students who wants to explore the theories,
Really great work 👏👏🙏🏻
Thanks Vivek!
Ya boi literally explained this in what took two weeks of lectures from my graduate classical mechanics professor. Very nice!
From a mathematical perspective the commutator is famous in the context of Lie Algebras, an example of which is first order differential operators. Who knew that math was useful in physics?
An absolutely crystal clear explanation of the subject. It did not remove the "weirdness" from QM as was not promised though!
He can't promise something that he is not capable of.
This video was amazing, thank you! Would love more higher level content from you :)
Thanks Abhay!
Cristal clear, thanks! Please post the Hamiltonian version of Noether's theorem!
Thanks Nicolas!
Amazing video!
This channel can be a perfect place for Screen Cleaning Wipes adds :)
I can not help myself, keep cleaning the screen while watching these wonderful videos.
Simply awesome.
Cool! Yes, please continue the subject and go towards Noether's theorem. Great job
Thanks Jarogniew!
This is Wonderful! I could have used this a week ago since our finals were just this week haha. You could also say "See this before you get to the end of e.g. Ch 13 of Taylor's Classical Mechanics!" :). I wish these were around when I got my undergrad degree in physics! Thank you! btw a video on Noether's Theorem in the Hamiltonian would be terrific!
Thanks! Better late than never!
Great explanation 😍😍😍😍
I agree with all the praises in the other comments, but I think it is understadted how excellent is your choice of topics. So far in the videos you made, you covered all the topics that I struggled to understand while trying to self-teach myself physics (mostly cause, in case of this concept and same goes for Lagrangian - they dont come of intutive concepts that map 1-1 to everyday physical quantities, I guess). Knowing to explain something in a clear manner is a skill indeed, but understanding what particular things other people might often not understand is a very important one as well.
Thanks Amir! Very glad they've been helpful!
ohh man I loove these videos so much!
#suggestion : Schrodinger equation. Explaining all that it contains without losing the mathematical essence. Thank you
Hi Elliot.
Do you have any suggestions about what type of physics should I learn and what playlists should I watch before starting QM?
I have an Issue with symbols and notations in QM.
Thanks a lot!
Very good video. I have a BSc in mathematical physics but end up doing quant finance... Ten years later, trying to re-read texts like Arnold's and Goldstein's Classical Mechanics is a daunting task but this kind of video makes it easier to regain intuition of the field. As a suggestion for the development of your channel, stick with relatively advanced stuff. I would definitely appreciate more videos on the geometric and symplectic interpretation of classical / quantum mechanics. Perhaps a video on the Dirac equation and/or particle physics would be cool as well.
Thanks Thierry!
Today I learned that the canonical commutation relation can be derived from the definition of the cross product in terms of the clifford product of the position and momentum... It was mind blowing.
Hi Elliot your videos are amazing and it would be really fantastic if you could do some videos on quantum filed theory.
Thanks Yair!
Do more about quantum mechanics basics please, Elliott!
Stay tuned AK!
Excellent!
Please more on the Hamiltonian Noether's Theorem
Sir, could you explain the role of Planck constant in Heisenberg uncertainty principle.
Would love to see the Hamiltonian version of Noether's theorem
Really great video! I would love to see a video on the Hamiltonian version of Noether's Theorem :)
Thanks Nick!
Yes please do explain Noethers Theorem and work it out with Hamiltonian formalism.
Already posted!
It was soo helpful
please make a video on Hamiltons version of noethers theorem. it sounds awesome
Great explanation as always!
Can you come up with a simiar video on Maxwell's Thermodynamic Relations or Quantum Stat or Ensemble Theory of Macroscopic systems?
Thanks Sachleen! I will think about your suggestions!
thank you for such a great video
Glad you liked it Shraddha!
Elliot, This video is excellent!
Do you a Math book list to prepare for studying of Quantum Mechanic?
Before you study quantum mechanics, take an introductory course in atomic physics. It will reduce your stress levels greatly.
Poisson bracket is wonderful bridge which can connect the clasiscal physics and quantum physics 😀
Superb!
This is very cool. Are they are relationships with the lagrangian or other variables in the poisson bracket? for example Q1 = x, Q2 = L?
Nothing interesting that comes to mind-you could look at Poisson brackets with the function p^2/2m - U, but I don’t know of any nice relations that result
very very good!
This is a great Video
In future videos, could you prevent the background from moving? It's way to fast if you are watching this video high.
Otherwise is a super interesting video! I learned a bunch of relations I didn't know before.
Wow, love at first sight with this channel. I discovered it because of Eigenchris. 😍
For the suggestion, can you please make more vids on String Theory. Thanks.
Thanks Mishti! Glad you liked it!
Please explain more about Noether’s theorem and its Hamiltonian versus Lagrangian versions
Made two videos about Noether's theorem, check them out!
Would love to see a video of Von Neumann's No hidden variables theorem since it is not covered in any undergraduate QM book. The theorem was very controversial and led to John Bell's theorem. Practically no videos on youtube about it.
Excellent
great video
Beautiful video! What software did you use to make it?
Thanks Joseph! This one I made mostly in Keynote.
yea pls.. do the hamiltonian version of noether also..
please cover Hamiltonian Noethr Thm. in another video. Thanks.
Sooooo sick!
Wow, criminally underrated. I had to stop and absorb some of the mathematics at times because it moved so fast, but I understood everything.
I really feel like my understanding of math and physics is leveling up.
Excellent! Absolutely, ideally I’d suggest going through it again after you watch and working through the equations yourself to make sure everything clicks
Hi elliot, thanks for another amezing video.now consider this following case is in realivistic physics when we derrive any quantity say velocity in x direction V_x, and it is constant so d V_x / dt = 0 ,. now in this case can we use poisson braecet ? like { V_x , H } = 0 ? but what about ' H ' ? im confused as in relativity tensors are included in general, so couldalong we have this poisson bracket with tensor ? or no need of this ? or simpli in relativity we cant use poisson bracket concept ?
Hi Pritam, in relativity the Hamiltonian for a free particle becomes H = \sqrt{p^2c^2 + m^2c^4}
@@PhysicswithElliot thank you elliot,... your video was a worthy for learning in deep level. I'm happy that many of my doubts been cleared by watching your video series.
what is this gem, wow!
please do cover the hamiltonian Noether theorem!
Wowww... I press the like button at the middle of video itself. This is gem... Enlightening. Kindly put some intuitive videos of role of matrices in quantum mechanics. For eg: representing spin as matrices (Pauli matrices) etc..etc. If possible kindly make one video on Hamilton -Jacobi theory and transition to quantum mechanics....
Thanks Deepak!
@@PhysicswithElliot Waiting for your new uploads.
h-bar is the reduced Planks constant.
h/2π = h-bar
h is Planks constant.
Can I know what tools you are using to create this content?
Why is partial derivative of momentum with respect to x = zero (and vice versa)? Momentum depends on velocity , which also depends on change in position x with respect to time.
They did not cover Lagrangians and Hamiltonians in my engineering education some 40+ years ago. But I understand that mathematics does two things: 1) Define new math, or 2) Derive new math. Newton had a physical model that he based his derivation of classical mechanics on. The gap in my understanding are the models behind the Lagrangians and Hamiltonians formulations of mechanics. They seem defined rather than derived and so appear arbitrary to me. For instance vectors have direction and magnitude and are easy to understand. But what does an "operator" have? Beats me. I would like to see some videos that fill in these gaps my education.
Def would have enjoyed this in undergrad
Hello sir for studying clasical mechanics or quantum mechanics from scratch how do you suggest to approach and pre requisites needed? Are both mechanics related or independent ? Thanks.
Multi-variate calculus, differential equations and linear algebra will suffice. If you want to be extra thorough, take an introductory class on functional analysis.
This is gold
Thanks Karan!
Nice video
It would be nice to have a relativistic version of this video.
Another video to add to your banger video streak
Just one question: how do you go from the classical {x ,p}=1 to the homologuous Qm 1/(ih_bar)[x_hat , p_hat] ? Maybe I lost focus somewhere because I only couldn't grasp that
Oh, lemme check the notes, I almost forgot about those! Maybe I can start to like homework now lol Your lessons are great reminders to me, but they also expand a bit on stuff I had already seen! Keep it goin
Thanks Arnaud! Oh I certainly didn’t explain why it works that way in going from classical to quantum mechanics, I only stated the rule without justification
Same question. I don't see the link between classical {x,p} and QM [x_hat,p_hat]. For instance, why there's a 1/(i*h_bar) factor? Maybe, I have to check the notes and the previous videos.
You can't give a proof or a demonstration of that transition, because QM is not contained in CM. Only a reasonable justification can be given. See chap IV of Dirac's principles of QM.
Extremely useful as always! Can you do some difficult examples? I usually find surface level explanation (as good as they may be), but I NEED deeper level explanation to help my understanding. Great content and thank you! Love from Tel Aviv
Thanks Dan!
Triple-A video. Thanks! Is "diagonalization" an option for a new video?
Thanks Jacob! And maybe!
[x,p] should be equal to 0 at classical limit! Since at the classical limit, the Planck constant \hbar goes to 0!
Hi, I am 3rd year of high school in already thinking about my future. I have found my self pretty interested in quantum physics or quantum mechanics as some like to call it. What would be your suggestion on where to start and what colleges I should consider. I go to ITHS information technology high school, I've had a brief moment with quantum physics.
Keep at it! Physics majors usually start taking quantum mechanics classes in their sophomore or junior year. In the meantime you can self study!
the possion bracket is more like a derivative because the jacobi identity is pretty much a product rule
I wonder if Poisson brackets have a matrix notation, allowing an easy conversion to matrix mechanics
Loved the bear =)
Also very nice video in general.
:)